Hindawi Publishing CorporationInternational Journal of Differential EquationsVolume 2013 Article ID 314290 11 pageshttpdxdoiorg1011552013314290

Research ArticleEmbedded Zassenhaus Expansion to Splitting SchemesTheory and Multiphysics Applications

Juumlrgen Geiser

Institute of Physics Felix-Hausdorff-Street 6 17489 Greifswald Germany

Correspondence should be addressed to Jurgen Geiser geisermathematikhu-berlinde

Received 25 March 2013 Revised 5 August 2013 Accepted 19 August 2013

Academic Editor Shuyu Sun

Copyright copy 2013 Jurgen GeiserThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We present some operator splitting methods improved by the use of the Zassenhaus product and designed for applications tomultiphysics problems We treat iterative splitting methods that can be improved by means of the Zassenhaus product formulawhich is a sequential splitting schemeThemain idea for reducing the computation time needed by the iterative scheme is to embedfast and cheap Zassenhaus product schemes since the computation of the commutators involved is very cheap since we are dealingwith nilpotent matrices We discuss the coupling ideas of iterative and sequential splitting techniques and their convergence Whilethe iterative splitting schemes converge slowly in their first iterative steps we improve the initial convergence rates by embeddingthe Zassenhaus product formula The applications are to multiphysics problems in fluid dynamics We consider phase models incomputational fluid dynamics and analyse how to obtain higher order operator splittingmethods based on the Zassenhaus productThe computational benefits derive from the use of sparse matrices which arise from the spatial discretisation of the underlyingpartial differential equations Since the Zassenhaus formula requires nearly constant CPU time due to its sparse commutators wehave accelerated the iterative splitting schemes

1 Introduction

Our motivation to study the operator splitting methodscomes from models in fluid dynamics for example problemsin bioremediation [1] or radioactive contaminants [2] Suchmultiphysics problems are delicate and the solver methodscan be accelerated by decoupling the different physicalbehaviours see [3 4] Based on the splitting error of all thesplitting schemes we have to take into account higher orderideas see [5 6] While standard splitting methods deal withlower order convergence see Lie splitting and Strang splitting[7 8] we propose a combination of iterative splittingmethodswith an embedded Zassenhaus product formula Such a com-bination allows reducing the splitting error and also reducingtheCPU time needed see [9]Theoretically we combine fixedpoint schemes (iterative splitting methods) with sequentialsplitting schemes (Zassenhaus products) which are con-nected with the theory of Lie groups and Lie algebras Basedon that relation we can construct higher order splittingschemes for an underlying Lie algebra and improve theconvergence results with cheap iterative schemes

Historically the efficiency of decoupling different physi-cal processes into simpler processes for example convectionand reaction has helped to accelerate the solution process andis discussed in [10]

We propose the following ideas

(i) Iterative splitting schemes are based on fixed pointschemes for example waveform relaxation whichlinearly improve the convergence order Based onreducing the integral operators to cheaply computablematrices they can be seen as solver methods see [9]

(ii) Zassenhaus formula is based on nested commutatorswhich are themain keys to deriving higher order stan-dard splitting schemes (eg Lie-Trotter and Strangsplitting)They are simple to compute because of theirnilpotent structure see [11]

In this paper we study the following ordinary differen-tial equations that can arose of semidiscretized partialdifferential equations for example multiphysics problem see[2 3] or theoretical physics see [12 13]

2 International Journal of Differential Equations

We focus our attention on the case of two linear operators(ie we consider the Cauchy problem) as follows

120597119888 (119905)

120597119905= 119860119888 (119905) + 119861119888 (119905) with 119905 isin [0 119879] 119888 (0) = 119888

0

(1)

whereby the initial function 1198880is given and 119860 and 119861 are

assumed to be bounded linear operators in the Banach spaceX with 119860 119861 X rarr X In realistic applications the operatorscorrespond to discretised matrices for example such asconvection and diffusion matrices see [2] Further they canbe givenmatrices for example such as in perturbation theory[14]

For all such problems the solution of the Cauchy problem(1) is given as

119888 (119905) = exp (119860 + 119861) 119888 (0) with 119905 isin [0 119879] (2)

where 119888(0) is the initial conditionHere one of the main ideas to compute such delicate

solutions is to separate or decouple the term exp(119860 + 119861) intosimpler and parallel computational expression for exampleexp(119860) exp(119861) see [10 15] To reduce the splitting error forsuch decompositions iterative splittingmethod and productsof exponential of noncommuting operators see [16 17] areimportantWe concentrate on iterative splittingmethods andimprove their convergence order with embedded Zassenhausexpansion see [18]

While iterative splitting methods are cheap to implementand can be computed fast one of their drawbacks is dueto their initial process Here Zassenhaus expansion is anattractive tool to overcome such a drawback and improvethe initial solutions to higher accuracy While on the otherhand we deal only with lower nested commutators of theZassenhaus formula the drawbacks of stringent regularityconditions on the operators are less see [19 20]

The outline of the present paper is as followsThe splittingmethods are discussed in Section 2 In Section 3 we presentthe numerical experiments and the benefits of the higherorder splittingmethods Finally we discuss future work in thearea of iterative and noniterative splitting methods

2 Splitting Methods

We consider the following splitting schemes

(i) iterative splitting schemes

(ii) Zassenhaus formula

Their ideas and their underlying convergence analysis basedon the Cauchy problem (1) are discussed in the next subsec-tions

21 Iterative Operator Splitting In the following we developtwo ideas of the iterative splitting schemes

Iterative splitting with respect to one operator (also calleda one-sided scheme) as follows

120597119888119894(119905)

120597119905= 119860119888119894(119905) + 119861119888

119894minus1(119905) with 119888

119894(119905119899) = 119888119899

119894 = 1 2 119898

(3)

Iterative splitting with respect to two operators (also called atwo-sided scheme) as follows

120597119888119894 (119905)

120597119905= 119860119888119894 (119905) + 119861119888

119894minus1 (119905) with 119888119894(119905119899) = 119888119899 (4)

120597119888119894+1 (119905)

120597119905= 119860119888119894(119905) + 119861119888

119894+1(119905) with 119888

119894+1(119905119899) = 119888119899

119894 = 1 3 2119898 + 1

(5)

Theorem 1 Let one considers the abstract Cauchy problemgiven in (1) One obtains for the one-sided iterative operatorsplitting method (4) the following accuracy

1003817100381710038171003817119878119894 minus exp ((119860 + 119861) 120591)1003817100381710038171003817 le 119862120591

119894 (6)

where 119878119894is the approximate solution for the 119894th iterative step

and 119862 is a constant that can be chosen uniformly on boundedtime intervals

Proof The proof is given for 119894 = 1 2 and with theconsistency error 119890

119894(120591) = 119888(120591) minus 119888

119894(120591) we have

119888119894(120591) = exp (119860120591) 119888 (119905

119899)

+ int

119905119899+1

119905119899

exp (119860 (119905119899+1

minus 119904)) 119861 exp (119904119860) 119888 (119905119899) 119889119904

+ int

119905119899+1

119905119899

exp (119860 (119905119899+1

minus 1199041)) 119861

times int

119905119899+1minus1199041

119905119899

exp ((119905119899+1

minus 1199041minus 1199042)119860)

times 119861 exp (1199042119860) 119888 (119905

119899) 11988911990421198891199041

+ sdot sdot sdot + int

119905119899+1

119905119899

exp (119860 (119905119899+1

minus 1199041)) 119861

sdot sdot sdot int

119905119899+1minussum119894minus2

119895=1119904119895

119905119899

exp((119905119899+1

minus

119894minus1

sum

119895=1

119904119895)119860)

times 119861119888init (119904119894) 119889119904119894119889119904119894minus1 11988911990421198891199041

119888 (120591) = exp (119860120591) 119888 (119905119899)

+ int

119905119899+1

119905119899

exp (119860 (119905119899+1

minus 119904)) 119861 exp (119904119860) 119888 (119905119899) 119889119904

+ int

119905119899+1

119905119899

exp (119860 (119905119899+1

minus 1199041)) 119861

times int

119905119899+1minus1199041

119905119899

exp ((119905119899+1

minus 1199041minus 1199042)119860)

times 119861 exp (1199042119860) 119888 (119905

119899) 11988911990421198891199041

International Journal of Differential Equations 3

+ sdot sdot sdot + int

119905119899+1

119905119899

exp (119860 (119905119899+1

minus 1199041)) 119861

sdot sdot sdot int

119905119899+1minussum119894minus1

119895=1119904119895

119905119899

exp((119905119899+1

minus

119894

sum

119895=1

119904119895)119860)

times 119861 exp ((119904119894(119860 + 119861)) 119888 (119905

119899) 119889119904119894 119889119904

21198891199041

(7)

We obtain1003817100381710038171003817119890119894

1003817100381710038171003817 le1003817100381710038171003817exp ((119860 + 119861) 120591) 119888 (119905

119899) minus 119878119894 (120591) 119888init (120591)

1003817100381710038171003817

le 119862120591119894 max119904119894isin[0120591]

1003817100381710038171003817exp (119904119894(119860 + 119861)) 119888 (119905

119899) minus 119888init (119904119894)

1003817100381710038171003817

le 119862120591119894 1003817100381710038171003817exp (120591 (119860 + 119861)) 119888 (119905

119899) minus 119888init (120591)

1003817100381710038171003817

(8)

where 119894 is the number of iterative stepsThe same idea can be applied to the even iterative scheme

and also for alternating 119860 and 119861

Remark 2 The accuracy of the initialisation 119888init(120591) is impor-tant for conserving or improving the underlying iterativesplitting scheme

Here we have the following initialisation schemes

1198881(120591) = exp (119860120591) 119888 (119905

119899) 997888rarr

10038171003817100381710038171198901198941003817100381710038171003817 le 119862120591

119894119888 (119905119899)

1198881(120591) = exp (119860120591) exp (119861120591) 119888 (119905

119899) 997888rarr

10038171003817100381710038171198901198941003817100381710038171003817 le 119862120591

119894+1119888 (119905119899)

(9)

So the slow convergence in the initial process 1198881(120591) of the

iterative splitting schemes can be improved by a cheapcomputable initial process 119888

1(120591) asymp exp((119860 + 119861)120591) We will

next discuss the fast convergent Zassenhaus formula for theinitial process

22 Zassenhaus Formula (Sequential Splitting) The Zassen-haus formula is an extension to the exponential splittingschemes and can be given inTheorem 3 see also [20]

Theorem 3 One assumes L(119860 119861) is the free Lie algebragenerated by 119860 and 119861 The kernel function of solution (1)exp((119860 + 119861)119905) can be uniquely decomposed as

exp ((119860 + 119861) 119905) = exp (119860119905) exp (119861119905) 120587119895

119894=1120587119898

119896=2

times exp (119862119896 (119860 119861) 119905

119896)

+ O (119905119898+1

)

(10)

where 119862119895(119860 119861) isin L(119860 119861) is a homogeneous Lie polynomial

of 119860 and 119861 of degree 119895

Proof The sketch of the proof is given in [20] The ideais based on the existence of the BCH (Baker-Campbell-Hausdorff) formula and we obtain

exp (minus119860119905) exp ((119860 + 119861) 119905) = exp (119861119905 + 119863 (119905)) (11)

where119863(119905) is a Lie polynomial of degree gt1 and

exp ((119860 + 119861) 119905) = exp (119860119905) exp (119861119905) + O (1199052) (12)

Further we have

exp (minus119861119905) exp (119861119905 + 119863 (119905)) = exp (11986221199052+ 119863 (119905)) (13)

where119863(119905) is a Lie polynomial of degree gt2 and

exp ((119860 + 119861) 119905) = exp (119860119905) exp (119861119905) exp (11986221199052) + O (119905

3)

(14)

We can repeat the recursive process and obtain by completeinduction the results

The first Zassenhaus polynomials are given as

1198622(119860 119861) =

1

2[119861 119860]

1198623 (119860 119861) =

1

3[[119861 119860] 119861] +

1

6[[119861 119860] 119860]

1198624 (119860 119861) =

1

24[[[119861 119860] 119860] 119860] +

1

8[[[119861 119860] 119861] 119860]

+1

8[[[119861 119860] 119861] 119861]

(15)

Here we see the benefit of such schemes in the computationof the commutators If we assume that we have quicklycomputable commutators for example nilpotent matricesor sparse matrices the computation time needed for theadjacent (or perturbed) operators exp(119862

119896119905119896) of the scheme is

much less than that of the main operators exp(119860119905) exp(119861119905)and so the scheme is very effective and can be used as aninitial solution of the iterative schemes

Remark 4 We can also generalize the application of theZassenhaus formula to decompositionmethods with existingBCH formulas for example

(i) 119860-119861 splitting or Lie-Trotter splitting(ii) Strang splitting

Example 5 (1)A-B or Lie-Trotter Splitting For the Lie-Trottersplitting there exists a BCH formula andwe have derived thepolynomials in Theorem 3

(2) Strang SplittingThere exists also a BCH formula which isgiven as

exp (119860119905

2) exp (119861119905) exp (

119860119905

2)= exp (119905119878

1+ 11990531198783+ 11990551198785+sdot sdot sdot )

(16)

where the 119878119894(119860 119861) isin L(119860 119861) is a homogeneous Lie

polynomial in 119860 and 119861 of degree 119894 see [21]The BCH formula is given as

exp ((119860

2+

119861

2) 119905) = Π

infin

119896=2exp (119862

119896119905119896) exp (

119860

2119905) exp(

119861

2119905)

exp((119861

2+

119860

2) 119905) = exp (

119861

2119905) exp(

119860

2119905)Πinfin

119896=2exp (119862

119896119905119896)

(17)

4 International Journal of Differential Equations

and so there exists a new product based on the Zassenhauspolynomials as follows

Πinfin

119896=3exp (119863

119896119905119896) = Π

infin

119896=2exp (119862

119896119905119896)Πinfin

119896=2exp (119862

119896119905119896) (18)

and we obtain a higher order scheme see also [22]

Remark 6 The application of the Zassenhaus formula topartial differential equations for example second-orderparabolic evolution equations needs additional stringentregularity conditions on the operators Due to the nestedcommutators which are needed to apply the Zassenhausformula one needs at least higher regularity conditions

We need some functional analytical results on thedomains of the operators The domains of the operators haveto satisfy the following compatibility conditions for the firstZassenhaus exponents

(i) Condition for the first term 1198622of the Zassenhaus

formula

D (1198712) sub D (119860

2) capD (119860119861) capD (119861119860) capD (119861

2) (19)

where119871 = 119860+119861 andD is the domain of the operators(ii) Condition for the second term 119862

3of the Zassenhaus

formula

D (1198713) sub D (119860

3) capD (119860

2119861)

capD (119860119861119860) capD (1198601198612) capD (119861

3)

(20)

where119871 = 119860+119861 andD is the domain of the operators

Example 7 If we assume to apply the Zassenhaus formula to asecond-order diffusion equation and we assume to split eachspatial direction see the functional analytical framework[23] where the domains are given as

D (119871) = 1198672(Ω) cap 119867

1

0(Ω)

D (119860) = 119888 isin 1198712(Ω) 120597

119909119909119888 120597119909119888 isin 1198712Ω

119906 (0 119910) = 119906 (1 119910) = 0 forall119910 isin (0 1)

D (119861) = 119888 isin 1198712(Ω) 120597

119910119910119888 120597119910119888 isin 1198712Ω

119906 (119909 0) = 119906 (119909 1) = 0 forall119909 isin (0 1)

(21)

whereΩ = (0 1)2

Then we need the following regularity result

(i) condition for the first term 1198622of the Zassenhaus

formula

D (1198712) sub 119867

4(Ω) (22)

(ii) condition for the second term 1198623of the Zassenhaus

formula

D (1198713) sub 119867

6(Ω) (23)

In the example we need a very high regular domain toapply a first or second Zassenhaus exponent This is one ofthe limitations and drawbacks of applying the Zassenhausformula In our applications we restrict on applying the firstor at least the second Zassenhaus exponent We apply thethird and fourth Zassenhaus exponents to the benchmarkproblems given as matrix equations without such restrictionsto the domains see [20]

23 Embedding the Zassenhaus Expansion into an IterativeSplitting Scheme We now discuss the embedding of theZassenhaus formula into the iterative operator splittingschemes

Definition 8 We apply the Lie-Trotter splitting with theembedded Zassenhaus formula which is given as

119864ZassenComp1 (119905) = exp (119860119905) exp (119861119905)

119864ZassenComp119895 (119905) = exp (119862119895119905119895) for 119895 isin 2 119894

(24)

where the sequential Zassenhaus operators are defined as

119864Zassen119894 (119905) = Π119894

119895=1119864ZassenComp119895 (119905) (25)

and we obtain an accuracy of O(119905119894) and 119894 is the number of

Zassenhaus components

In the following theorem we present the improvement ofthe iterative splitting scheme with the Zassenhaus expansion

Theorem 9 One solves the initial value problem (1) Weassume bounded and constant operators 119860 119861

The initialisation process is done with the Zassenhausformula

119888119894(119905) = 119864

119885119886119904119904119890119899119894(119905) 1198880 (26)

where 119864119885119886119904119904119890119899119894

(119905) is given in (25)Furthermore the improved solutions are embedded in the

iterative splitting scheme (4) and one has after 119895 iterative stepsthe following result

119888119894+119895

(119905) = 119864119894119905119890119903119895

(119905) 119864119885119886119904119904119890119899119894

(119905) 1198880 (27)

This has improved the error of the iterative scheme to O(119905119894+119895

)

Proof The solution of the iterative splitting scheme (4) is

119888119894+119895

(119905) = 119864iter119894 (119905) 119888init119895 (28)

where 119878119894(119905) = 119864iter119894(119905) given inTheorem 1

The initialisation is given by the Zassenhaus formula

119888init119895 (119905) = 119864Zassen119895 (119905) 1198880 (29)

Combining both splitting schemes we have a local error of

119890119894+119895 (119905) =

10038171003817100381710038171003817119888 (119905) minus 119888

119894+119895 (119905)10038171003817100381710038171003817

=10038171003817100381710038171003817119888 (119905) minus 119864iter119894 (119905) 119864Zassen119895 (119905) 1198880

10038171003817100381710038171003817

le O (119905119894+119895

) 1198880

(30)

International Journal of Differential Equations 5

Remark 10 We obtain an acceleration of the iterative schemeby applying cheap higher order Zassenhaus formula Such abenefit allows reducing to 2-3 iterative steps while the initialsolution is of a higher order Taking into account the sparsityor nilpotency of the commutators we nowhave a fast iterativesplitting scheme

Remark 11 For our numerical experiment we apply theZassenhaus formulaat the beginning of each integration stepas the so-called preprocessor Therefore we obtain moreaccurate starting conditions for the integration steps of theiterative splitting scheme Based on the regularity conditionsof the nested commutator we apply only 1-2 Zassenhausexponents

In the following we explain the final algorithm ofthe embedded Zassenhaus formula to the iterative splittingscheme

Algorithm 12 We compute 119873 time steps where each timestep Δ119905

119899is given as Δ119905

119899= 119905119899+1

minus 119905119899 with 119899 = 0 119873 minus 1

Further 119888(1199050) is the initial condition of the Cauchy problem

(1) and we compute 119888(1199051) 119888(119905119873) successively We have thefollowing two steps first we apply the Zassenhaus formulaas initialization of 119888

0(119905119899+1

) for the iterative scheme secondwe apply the iterative splitting scheme to the initial valueproblem (1) and compute the final solution 119888(119905

119899+1) as follows

(1) We start with 119895 = 2 and compute the Zassenhausformula

(2) We compute the matrix norm for the Zassenhaussexponents 119862

119895

If errZass119895

le err1 then we start with the iterative

splitting scheme with 1198880(119905) = 119888Zass

119895

119894 = 1 and go to(3) else 119895 = 119895 + 1 and we go to (1)

(3) If the error of the iterative scheme erriter le err2 then

we are finished else 119894 = 119894+1 and we compute the nextiterative step till erriter le err

2

3 Numerical Examples

In this section we discuss examples of the use of embeddingZassenhaus product methods in iterative splitting schemesIn the first example we demonstrate somewhat artificiallyhow the proposed Zassenhaus splitting method avoids thesplitting error up to a certain order The next example showsthe benefit of the combined splitting schemes for applicationsto multiphysics problems

In the following we consider a benchmark example tovalidate the theoretical results and real life problems to applyour schemes

We applied the following splitting schemes given inTable 1

31 Benchmark Problem Matrix Examples In the followingthe idea of thematrix example is to present the decompositioninto two simpler matrices In context of the multiphysics

Table 1 Applied splitting schemes

Abbreviation inthe examples Splitting methods

A-B A-B splittingStrang Strang splitting1198881

Iterative splitting with one iterative step

1198882

Iterative splitting with one iterative step and oneZassenhaus step as prestep

1198883

Iterative splitting with one iterative step and twoZassenhaus steps as presteps

1198884

Iterative splitting with two iterative steps andtwo Zassenhaus steps as presteps

1198885

Iterative splitting with three iterative steps andtwo Zassenhaus steps as presteps

1198886

Iterative splitting with four iterative steps andtwo Zassenhaus steps as presteps

problems even such simple problems can be seen as testexamples We assume that each simple matrix is presenting abasic single physics problems for example reaction processbetween two species while the combination of the twomatrices results in a more complicate physics problem forexample coupled reaction process between two species see[2]

We consider the matrix equation

1199061015840(119905) = [

1 2

3 0] 119906 119906 (0) = 119906

0= (

0

1) 119905 isin [0 1] (31)

the exact solution of which is

119906 (119905) =2 (1198903119905

minus 119890minus2119905

)

5 (32)

We split the matrix as

[1 2

3 0] = [

1 1

1 0] + [

0 1

2 0] = 119860 + 119861 (33)

where [119860 119861] = 0 For this simple example we did not obtainnilpotent matrices for the commutators

We apply 10ndash100 time steps for the computations Thefinal integration time of each single time step was less thanlt1 [sec] such that we need at least for such benchmarkproblems about 1ndash10 [sec] Figure 1 presents the numericalerrors that is the difference between the exact and thenumerical solution

Figure 2 presents the CPU times needed by the standardand the iterative splitting schemes The application of theZassenhaus formula did not increase while the next Zassen-haus coefficients are at least nearly nilpotent matrices and wecan neglect the computational timeMoreover different time-steps did not influence the computations of the Zassenhausexponents which is a preprocess First we compute and saveall the Zassenhaus exponents and later we multiply with thenecessary time steps

6 International Journal of Differential Equations

102

102

10minus2

10minus4

10minus6

10minus8

10minus2

10minus1

100

AB Strang one-sided

AB

Δt

Strang

c1

c2

c3

c4

c5

c6

Erro

r L1

(a)

AB

Δt

Strang

102

102

10minus2

10minus4

10minus6

10minus8

10minus2

10minus1

100

AB Strang two-sided

c1

c2

c3

c4

c5

c6

Erro

r L1

(b)

Figure 1 Numerical errors of the standard splitting scheme and theiterative schemes with the Zassenhaus formula where we have atleast 1 4 iterative steps

Remark 13 We see that the errors decrease significantlywith increasing order of approximation for the initialisa-tion process from the Zassenhaus splitting (we apply 1ndash3Zassenhaus coefficients) Here we apply commutators whichdid not reduce their matrix rank therefore we have thesame computation time as for the pure iterative schemesOverall we have their benefits in the application of theZassenhaus product schemes to the standard Lie-Trotter orStrang-Marchuk splitting Similar results are given withmoreiterative steps 119894 = 3 6 as expected

32 Real Life Problems Multiphase Examples In the nextexamples we simulate coupled transport and reaction equa-tions with different phases so-called multiphase problems

CPU

tim

e

AB Strang one-sided

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

(a)

CPU

tim

e

AB Strang two-sided

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

(b)

Figure 2 CPU times of the standard splitting scheme and theiterative schemes with the Zassenhaus formula where we have atleast 1 4 iterative steps

see [9]The equations are coupledwith the reaction terms andare presented as follows

120597119905119877119894119906119894+ nabla sdot v119906

119894= minus 120582

119894119877119894119906119894+ 120582119894minus1

119877119894minus1

119906119894minus1

+ 120573 (minus119906119894+ 119892119894) inΩ times (0 119879)

1199061198940

(119909) = 119906119894(119909 0) on Ω

120597119905119877119894119892119894= minus 120582

119894119877119894119892119894+ 120582119894minus1

119877119894minus1

119892119894minus1

+ 120573 (minus119892119894+ 119906119894) in Ω times (0 119879)

1198921198940

(119909) = 119892119894(119909 0) on Ω

119894 = 1 119898

(34)

International Journal of Differential Equations 7

where 119898 is the number of equations and 119894 is the index ofeach component The unknown mobile concentrations 119906

119894=

119906119894(119909 119905) are taken in Ω times (0 119879) sub R119899 times R+ where 119899 is the

spatial dimension The unknown immobile concentrations119892119894= 119892119894(119909 119905) are taken in Ω times (0 119879) sub R119899 times R+ where 119899 is

the spatial dimensionThe retardation factors119877119894are constant

and 119877119894

ge 0 The kinetic part is given by the factors 120582119894

They are constant and 120582119894ge 0 For the initialisation of the

kinetic part we set 1205820

= 0 The kinetic part is linear andirreversible so the successors have only one predecessor Theinitial conditions are given for each component 119894 as constantsor linear impulses For the boundary conditions we havetrivial inflow and outflow conditions with 119906

119894= 0 at the inflow

boundary The transport part is given by the velocity v isin R119899

and is piecewise constant see [2] The exchange between themobile and immobile part is given by 120573

In the following we discuss the one-phase and two-phaseexamples

321 One-Phase Example The next example is a simplifiedreal life problem with a multiphase transport-reaction equa-tion We treat the mobile and immobile pores in a porousmedia Such simulations are made about waste scenarios see[2 24]

We concentrate on the computational benefits of a fastcomputation of the mixed iterative scheme with the Zassen-haus formula

The one phase equation is

1205971199051198881+ nabla sdot F

11198881= minus12058211198881 in Ω times [0 119905]

1205971199051198882+ nabla sdot F

21198882= 12058211198881minus 12058221198882 in Ω times [0 119905]

F119894= v119894minus 119863119894nabla 119894 = 1 2

1198881(x 119905) = 119888

10(x) 119888

2(x 119905) = 119888

20(x) on Ω

1198881 (x 119905) = 119888

11 (x 119905) 1198882 (x 119905) = 119888

21 (x 119905)

on 120597Ω times [0 119905]

(35)

Here the parameters take the values V1= 01 V

2= 005119863

1=

0011198632= 0005 and 120582

1= 1205822= 01

We now turn to the finite difference schemes andsemidiscretise the equation with its convection and diffusionoperators as follows

120597119905c = (119860

1+ 1198602) c (36)

We obtain the two matrices and decouple the diffusion andconvection parts as follows

1198601= (

119860diff 0

0 119860diff) isin R

2119868times2119868

1198602= (

119860Conv 0

0 119860Conv) + (

minusΛ1

0

Λ1

minusΛ2

) isin R2119868times2119868

(37)

We apply the splitting method to the operators 1198601and 119860

2

given in Section 21

The submatrices are

119860diff119894 =119863119894

Δ11990921198771=

119863119894

Δ1199092sdot (

minus2 1

1 minus2 1

d d d1 minus2 1

1 minus2

) isin R119868times119868

119860conv119894 = minusV119894

Δ1199091198772= minus

V119894

Δ119909sdot (

1

minus1 1

d dminus1 1

minus1 1

) isin R119868times119868

(38)

where 119868 is the number of spatial points and Δ119909 is the spatialstep size Consider

Λ1= (

1205821

0

0 1205821

0

d d d0 1205821

0

0 1205821

) isin R119868times119868

Λ2= (

1205822

0

0 1205822

0

d d d0 1205822

0

0 1205822

) isin R119868times119868

(39)

We have the commutator

[1198601 1198602] = (

0 0

1198632minus 1198631

Δ11990921198771Λ1

0) isin R

2119868times2119868 (40)

where we assume that [1198771 1198772] asymp 0 This is a nilpotent

commutator and the computation time needed for thecommutators of such operators is less than that for the fulloperators Furthermore we assume that we have a boundedspatial step size which is given by Δ119909 = 01 ≫ 0 so thatwe scale the singular entry (119863

2minus 1198631)Δ1199092asymp 1 and evade a

blowup in such matricesWe have obtained the optimal results of a basically

constant CPU time for decreasing time steps which increasein the standard scheme

We apply 10ndash100 timesteps for the computations with 10ndash100 spatial points The final integration time of each singletime step was about 1ndash10 [sec] such that we need at least forsuch benchmark problems about 100ndash1000 [sec]

Figure 3 presents the numerical error that is the differ-ence between the exact and the numerical solution

Figure 4 presents the CPU time of the standard andthe iterative splitting schemes Based on the preprocess ofcomputing the Zassenhaus exponents at the beginning andstore the matrices we could save additional computationaltime with larger time steps

Remark 14 For the iterative schemes with embedded Zassen-haus products we can reach improved results more quicklyThe benefits come from the constant CPU time needed for

8 International Journal of Differential Equations

AB

Δt

Strang

101

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

Erro

r L1

AB Strang one-sided with A

(a)

AB

Δt

Strang

102

100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

Erro

r L1

AB Strang one-sided with B

(b)

AB

Δt

Strang

102

100

10minus2

10minus4

10minus6

10minus8

10minus2

10minus1

100

AB Strang two-side

c1

c2

c3

c4

c5

c6

Erro

r L1

(c)

Figure 3 Numerical errors of the standard splitting scheme and theiterative schemes with the Zassenhaus formula where we have atleast 1 4 iterative steps

CPU

tim

e

AB strang one-side with A

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

(a)

CPU

tim

e

Δt

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

AB

Strang

c1

c2

c3

c4

c5

c6

AB Strang one-sided with B

(b)

CPU

tim

e

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

AB Strang two-side

c1

c2

c3

c4

c5

c6

(c)

Figure 4 CPU time of the standard splitting scheme and theiterative schemes with the Zassenhaus formula where we have atleast 1 4 iterative steps

International Journal of Differential Equations 9

calculating the nilpotent commutators in the Zassenhausformula So it makes sense to have 1ndash3 Zassenhaus stepsbefore starting the iterative scheme At that point only 2-3iterative steps are needed to obtainmore accurate results thanthose from the expensive standard schemes Here too weobtain the best results with the one-sided iterative schemes

Remark 15 Based on our restriction to 1198631 1198632 and Δ119909 with

(1198632minus 1198631)Δ1199092

asymp 1 we can control the delicate blowupof the computed commutators By the way if we neglectthe restriction and Δ119909 with (119863

2minus 1198631)Δ1199092

rarr infin thecommutator blows up and we lose the benefits of such analgorithmThis is a limitation of applying such an embeddedZassenhaus formula

322 Two Phase Example The next example is a moredelicate real life problem involving a multiphase transport-reaction equation We treat mobile and immobile pores in aporousmedia Such simulations aremade forwaste scenariossee [2]

We concentrate on the computational benefits of a fastcomputation of the commutators and their use for theinitialisation of the iterative splitting scheme

The equation is

1205971199051198881+ nabla sdot F119888

1= 119892 (minus119888

1+ 1198881119894119898

) minus 12058211198881 in Ω times [0 119905]

1205971199051198882+ nabla sdot F119888

2= 119892 (minus119888

2+ 1198882119894119898

) + 12058211198881minus 12058221198882 in Ω times [0 119905]

F = v minus 119863nabla

1205971199051198881119894119898

= 119892 (1198881minus 1198881119894119898

) minus 12058211198881119894119898

in Ω times [0 119905]

1205971199051198882119894119898

= 119892 (1198882minus 1198882119894119898

) + 12058211198881119894119898

minus 12058221198882119894119898

in Ω times [0 119905]

1198881(x 119905) = 119888

10(x) 119888

2(x 119905) = 119888

20(x) on Ω

1198881(x 119905) = 119888

11(x 119905) 119888

2(x 119905) = 119888

21(x 119905)

on 120597Ω times [0 119905]

1198881119894119898

(x 119905) = 0 1198882119894119898

(x 119905) = 0 on Ω

1198881119894119898 (x 119905) = 0 119888

2119894119898 (x 119905) = 0 on 120597Ω times [0 119905]

(41)

Here the parameters are V = 01 119863 = 001 1205821= 1205822= 01

and 119892 = 001We next treat the semidiscretised equation given by the

matrices

120597119905C = (

119860 minus Λ1minus 119866 0 119866 0

Λ1

119860 minus Λ2minus 119866 0 119866

119866 0 minusΛ1minus 119866 0

0 119866 Λ1

minusΛ2minus 119866

)C

(42)

where C = (c1 c2 c1119894119898 c2119894119898)119879 while c1 = (119888

11 119888

1119868) is the

solution of the first species in themobile phase in each spatialdiscretisation point (119894 = 1 119868) and the same for the othersolution vectors

We have the following two operators for the splittingmethod

119860 =119863

Δ1199092sdot (

minus2 1

1 minus2 1

d d d1 minus2 1

1 minus2

)

+VΔ119909

sdot (

1

minus1 1

d dminus1 1

minus1 1

) isin R119868times119868

(43)

where 119868 is the number of spatial pointsConsider

Λ1= (

1205821

0

0 1205821

0

d d d0 1205821

0

0 1205821

)

Λ2= (

1205822

0

0 1205822

0

d d d0 1205822

0

0 1205822

) isin R119868times119868

119866 = (

119892 0

0 119892 0

d d d0 119892 0

0 119892

) isin R119868times119868

(44)

We decouple this into the following matrices

119860 = (

119860 0 0 0

0 119860 0 0

0 0 0 0

0 0 0 0

) isin R4119868times4119868

1198611= (

minusΛ1

0 0 0

Λ1

minusΛ2

0 0

0 0 minusΛ1

0

0 0 Λ1

minusΛ2

)

1198612= (

minus119866 0 119866 0

0 minus119866 0 119866

119866 0 minus119866 0

0 119866 0 minus119866

) isin R4119868times4119868

(45)

For the operators 119860 and 119861 = 1198611+ 1198612and 119861 we apply the

iterative splitting method given in Section 21Based on the decomposition119860 is block diagonal and 119861 is

tridiagonalThe commutator is

[119860 119861] = (

0 0 119860119866 0

0 0 0 119860119866

minus119860119866 0 0 0

0 minus119860119866 0 0

) isin R4119868times4119868

(46)

10 International Journal of Differential Equations

Strang

AB

102

101

100

10minus1

10minus2

10minus3

100

Δt

c1

c2

c3

c4

c5

c6

Erro

r L1

AB Strang one-sided with B

(a)

CPU

tim

e

AB Strang one-side

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

(b)

Figure 5 Numerical errors and CPU time of the standard splittingscheme and the iterative schemes with Zassenhaus formula wherewe have at least 1 4 iterative steps

where we have a sparser operator than the main operator 119861and therefore we save computation time in computing suchoperators Further we assume that we have a bounded spatialstep size which is given by Δ119909 = 01 ≫ 0 so that we scale thesingular entry119863Δ119909

2asymp 1 and VΔ119909 asymp 1

Figure 5 presents the numerical error that is the differ-ence between the exact and the numerical solution and theCPU time required The optimal results are from the one-sided iterative schemes on the operator 119861

Remark 16 We obtain the same results as in the one-phase example because of the basically constant CPUtime for decreasing time steps of the iterative scheme weobtain optimal results when applying 1ndash3 Zassenhaus stepsTherefore with the embedded Zassenhaus formula we can

reach improved results more quickly than with the standardschemes With 2-3 more iterative steps we obtain moreaccurate results With the one-sided iterative schemes wereach the best convergence results when using the moredelicate operator 119861

4 Conclusions and Discussion

In this paper we have presented a novel splitting schemecombining the ideas of iterative and sequential schemes basedon the Zassenhaus formula Here the idea is to decouplethe expensive iterative computation of the schemes basedonly on the matrix exponential obtaining simpler embeddedZassenhaus schemes based on nilpotent commutators Suchcombinations have the benefit of reducing the computationtime because the commutators can be computed very cheaplyOn the other hand simple linear iterative steps can be donevery cheaply if they are initialised with a sufficiently accuratestarting solution and this accelerates the solvers The erroranalysis showed that these are stablemethods for higher orderschemes In the applications we were able to demonstratethe speedup from the Zassenhaus enhanced method andwere able to apply it to multiphysics applications In thefuture we will concentrate on linear and nonlinear matrix-dependent schemes that switch between noniterative anditerative schemes based on Zassenhaus products

References

[1] R E Ewing ldquoUp-scaling of biological processes andmultiphaseow in porous mediardquo in IIMA Volumes in Mathematics andIts Applications vol 295 pp 195ndash215 Springer New York NYUSA 2002

[2] J Geiser Discretization and Simulation of Systems forConvection-Diffusion-Dispersion Reactions with Applicationsin Groundwater Contamination Groundwater ModellingManagement and Contamination Nova Science MonographNew York NY USA 2008

[3] N Antonic C J van Duijn W Jager and A MikelicMultiscaleProblems in Science and Technology Challenges toMathematicalAnalysis and Perspectives Springer Berlin Germany 2002

[4] E Weinan Principle of Multiscale Modelling Cambridge Uni-versity Press Cambridge UK 2011

[5] S Blanes and F Casas ldquoOn the necessity of negative coefficientsfor operator splitting schemes of order higher than twordquoAppliedNumerical Mathematics vol 54 no 1 pp 23ndash37 2005

[6] EHansen andAOstermann ldquoHigh order splittingmethods foranalytic semigroups existrdquo BIT NumericalMathematics vol 49no 3 pp 527ndash542 2009

[7] G Strang ldquoOn the construction and comparison of differenceschemesrdquo SIAM Journal on Numerical Analysis vol 5 pp 506ndash517 1968

[8] G I Marchuk ldquoSome applicatons of splitting-up methods tothe solution of problems in mathematical physicsrdquo AplikaceMatematiky vol 1 pp 103ndash132 1968

[9] J Geiser ldquoComputing exponential for iterative splitting meth-ods algorithms and applicationsrdquo Journal of AppliedMathemat-ics vol 2011 Article ID 193781 27 pages 2011

[10] R I McLachlan and G RW Quispel ldquoSplittingmethodsrdquoActaNumerica vol 11 pp 341ndash434 2002

International Journal of Differential Equations 11

[11] J Geiser G Tanoglu and N Gucuyenen ldquoHigher order oper-ator splitting methods via Zassenhaus product formula theoryand applicationsrdquo Computers amp Mathematics with Applicationsvol 62 no 4 pp 1994ndash2015 2011

[12] G H Weiss and A A Maradudin ldquoThe Baker-Hausdorff for-mula and a problem in crystal physicsrdquo Journal of MathematicalPhysics vol 3 pp 771ndash777 1962

[13] R MWilcox ldquoExponential operators and parameter differenti-ation in quantum physicsrdquo Journal of Mathematical Physics vol8 pp 962ndash982 1967

[14] J Geiser ldquoAn iterative splitting approach for linear integro-differential equationsrdquo Applied Mathematics Letters vol 26 no11 pp 1048ndash1052 2013

[15] F Casas and A Murua ldquoAn efficient algorithm for computingthe Baker-Campbell-Hausdorff series and some of its applica-tionsrdquo Journal of Mathematical Physics vol 50 no 3 Article ID033513 2009

[16] J Geiser Iterative Splitting Methods for Differential EquationsChapmanampHallCRCNumerical Analysis and Scientific Com-puting CRC Press Boca Raton Fla USA 2011

[17] S Vandewalle Parallel Multigrid Waveform Relaxation forParabolic Problems Teubner Stuttgart Germany 1993

[18] J Geiser and G Tanoglu ldquoOperator-splitting methods viathe Zassenhaus product formulardquo Applied Mathematics andComputation vol 217 no 9 pp 4557ndash4575 2011

[19] F Bayen ldquoOn the convergence of the Zassenhaus formulardquoLetters in Mathematical Physics vol 3 no 3 pp 161ndash167 1979

[20] F Casas A Murua and M Nadinic ldquoEfficient computation ofthe Zassenhaus formulardquo Computer Physics Communicationsvol 183 no 11 pp 2386ndash2391 2012

[21] E Hairer C Lubich and GWannerGeometric Numerical Inte-gration vol 31 of Springer Series in Computational MathematicsSpringer Berlin Germany 2002

[22] H Yoshida ldquoConstruction of higher order symplectic integra-torsrdquo Physics Letters A vol 150 no 5ndash7 pp 262ndash268 1990

[23] E Faou A Ostermann and K Schratz ldquoAnalysis of exponentialsplitting methods for inhomogeneous parabolic equationsrdquohttparxivorgabs12125827

[24] J Geiser ldquoDiscretization methods with embedded analyticalsolutions for convection-diffusion dispersion-reaction equa-tions and applicationsrdquo Journal of EngineeringMathematics vol57 no 1 pp 79ndash98 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 International Journal of Differential Equations

We focus our attention on the case of two linear operators(ie we consider the Cauchy problem) as follows

120597119888 (119905)

120597119905= 119860119888 (119905) + 119861119888 (119905) with 119905 isin [0 119879] 119888 (0) = 119888

0

(1)

whereby the initial function 1198880is given and 119860 and 119861 are

assumed to be bounded linear operators in the Banach spaceX with 119860 119861 X rarr X In realistic applications the operatorscorrespond to discretised matrices for example such asconvection and diffusion matrices see [2] Further they canbe givenmatrices for example such as in perturbation theory[14]

For all such problems the solution of the Cauchy problem(1) is given as

119888 (119905) = exp (119860 + 119861) 119888 (0) with 119905 isin [0 119879] (2)

where 119888(0) is the initial conditionHere one of the main ideas to compute such delicate

solutions is to separate or decouple the term exp(119860 + 119861) intosimpler and parallel computational expression for exampleexp(119860) exp(119861) see [10 15] To reduce the splitting error forsuch decompositions iterative splittingmethod and productsof exponential of noncommuting operators see [16 17] areimportantWe concentrate on iterative splittingmethods andimprove their convergence order with embedded Zassenhausexpansion see [18]

While iterative splitting methods are cheap to implementand can be computed fast one of their drawbacks is dueto their initial process Here Zassenhaus expansion is anattractive tool to overcome such a drawback and improvethe initial solutions to higher accuracy While on the otherhand we deal only with lower nested commutators of theZassenhaus formula the drawbacks of stringent regularityconditions on the operators are less see [19 20]

The outline of the present paper is as followsThe splittingmethods are discussed in Section 2 In Section 3 we presentthe numerical experiments and the benefits of the higherorder splittingmethods Finally we discuss future work in thearea of iterative and noniterative splitting methods

2 Splitting Methods

We consider the following splitting schemes

(i) iterative splitting schemes

(ii) Zassenhaus formula

Their ideas and their underlying convergence analysis basedon the Cauchy problem (1) are discussed in the next subsec-tions

21 Iterative Operator Splitting In the following we developtwo ideas of the iterative splitting schemes

Iterative splitting with respect to one operator (also calleda one-sided scheme) as follows

120597119888119894(119905)

120597119905= 119860119888119894(119905) + 119861119888

119894minus1(119905) with 119888

119894(119905119899) = 119888119899

119894 = 1 2 119898

(3)

Iterative splitting with respect to two operators (also called atwo-sided scheme) as follows

120597119888119894 (119905)

120597119905= 119860119888119894 (119905) + 119861119888

119894minus1 (119905) with 119888119894(119905119899) = 119888119899 (4)

120597119888119894+1 (119905)

120597119905= 119860119888119894(119905) + 119861119888

119894+1(119905) with 119888

119894+1(119905119899) = 119888119899

119894 = 1 3 2119898 + 1

(5)

Theorem 1 Let one considers the abstract Cauchy problemgiven in (1) One obtains for the one-sided iterative operatorsplitting method (4) the following accuracy

1003817100381710038171003817119878119894 minus exp ((119860 + 119861) 120591)1003817100381710038171003817 le 119862120591

119894 (6)

where 119878119894is the approximate solution for the 119894th iterative step

and 119862 is a constant that can be chosen uniformly on boundedtime intervals

Proof The proof is given for 119894 = 1 2 and with theconsistency error 119890

119894(120591) = 119888(120591) minus 119888

119894(120591) we have

119888119894(120591) = exp (119860120591) 119888 (119905

119899)

+ int

119905119899+1

119905119899

exp (119860 (119905119899+1

minus 119904)) 119861 exp (119904119860) 119888 (119905119899) 119889119904

+ int

119905119899+1

119905119899

exp (119860 (119905119899+1

minus 1199041)) 119861

times int

119905119899+1minus1199041

119905119899

exp ((119905119899+1

minus 1199041minus 1199042)119860)

times 119861 exp (1199042119860) 119888 (119905

119899) 11988911990421198891199041

+ sdot sdot sdot + int

119905119899+1

119905119899

exp (119860 (119905119899+1

minus 1199041)) 119861

sdot sdot sdot int

119905119899+1minussum119894minus2

119895=1119904119895

119905119899

exp((119905119899+1

minus

119894minus1

sum

119895=1

119904119895)119860)

times 119861119888init (119904119894) 119889119904119894119889119904119894minus1 11988911990421198891199041

119888 (120591) = exp (119860120591) 119888 (119905119899)

+ int

119905119899+1

119905119899

exp (119860 (119905119899+1

minus 119904)) 119861 exp (119904119860) 119888 (119905119899) 119889119904

+ int

119905119899+1

119905119899

exp (119860 (119905119899+1

minus 1199041)) 119861

times int

119905119899+1minus1199041

119905119899

exp ((119905119899+1

minus 1199041minus 1199042)119860)

times 119861 exp (1199042119860) 119888 (119905

119899) 11988911990421198891199041

International Journal of Differential Equations 3

+ sdot sdot sdot + int

119905119899+1

119905119899

exp (119860 (119905119899+1

minus 1199041)) 119861

sdot sdot sdot int

119905119899+1minussum119894minus1

119895=1119904119895

119905119899

exp((119905119899+1

minus

119894

sum

119895=1

119904119895)119860)

times 119861 exp ((119904119894(119860 + 119861)) 119888 (119905

119899) 119889119904119894 119889119904

21198891199041

(7)

We obtain1003817100381710038171003817119890119894

1003817100381710038171003817 le1003817100381710038171003817exp ((119860 + 119861) 120591) 119888 (119905

119899) minus 119878119894 (120591) 119888init (120591)

1003817100381710038171003817

le 119862120591119894 max119904119894isin[0120591]

1003817100381710038171003817exp (119904119894(119860 + 119861)) 119888 (119905

119899) minus 119888init (119904119894)

1003817100381710038171003817

le 119862120591119894 1003817100381710038171003817exp (120591 (119860 + 119861)) 119888 (119905

119899) minus 119888init (120591)

1003817100381710038171003817

(8)

where 119894 is the number of iterative stepsThe same idea can be applied to the even iterative scheme

and also for alternating 119860 and 119861

Remark 2 The accuracy of the initialisation 119888init(120591) is impor-tant for conserving or improving the underlying iterativesplitting scheme

Here we have the following initialisation schemes

1198881(120591) = exp (119860120591) 119888 (119905

119899) 997888rarr

10038171003817100381710038171198901198941003817100381710038171003817 le 119862120591

119894119888 (119905119899)

1198881(120591) = exp (119860120591) exp (119861120591) 119888 (119905

119899) 997888rarr

10038171003817100381710038171198901198941003817100381710038171003817 le 119862120591

119894+1119888 (119905119899)

(9)

So the slow convergence in the initial process 1198881(120591) of the

iterative splitting schemes can be improved by a cheapcomputable initial process 119888

1(120591) asymp exp((119860 + 119861)120591) We will

next discuss the fast convergent Zassenhaus formula for theinitial process

22 Zassenhaus Formula (Sequential Splitting) The Zassen-haus formula is an extension to the exponential splittingschemes and can be given inTheorem 3 see also [20]

Theorem 3 One assumes L(119860 119861) is the free Lie algebragenerated by 119860 and 119861 The kernel function of solution (1)exp((119860 + 119861)119905) can be uniquely decomposed as

exp ((119860 + 119861) 119905) = exp (119860119905) exp (119861119905) 120587119895

119894=1120587119898

119896=2

times exp (119862119896 (119860 119861) 119905

119896)

+ O (119905119898+1

)

(10)

where 119862119895(119860 119861) isin L(119860 119861) is a homogeneous Lie polynomial

of 119860 and 119861 of degree 119895

Proof The sketch of the proof is given in [20] The ideais based on the existence of the BCH (Baker-Campbell-Hausdorff) formula and we obtain

exp (minus119860119905) exp ((119860 + 119861) 119905) = exp (119861119905 + 119863 (119905)) (11)

where119863(119905) is a Lie polynomial of degree gt1 and

exp ((119860 + 119861) 119905) = exp (119860119905) exp (119861119905) + O (1199052) (12)

Further we have

exp (minus119861119905) exp (119861119905 + 119863 (119905)) = exp (11986221199052+ 119863 (119905)) (13)

where119863(119905) is a Lie polynomial of degree gt2 and

exp ((119860 + 119861) 119905) = exp (119860119905) exp (119861119905) exp (11986221199052) + O (119905

3)

(14)

We can repeat the recursive process and obtain by completeinduction the results

The first Zassenhaus polynomials are given as

1198622(119860 119861) =

1

2[119861 119860]

1198623 (119860 119861) =

1

3[[119861 119860] 119861] +

1

6[[119861 119860] 119860]

1198624 (119860 119861) =

1

24[[[119861 119860] 119860] 119860] +

1

8[[[119861 119860] 119861] 119860]

+1

8[[[119861 119860] 119861] 119861]

(15)

Here we see the benefit of such schemes in the computationof the commutators If we assume that we have quicklycomputable commutators for example nilpotent matricesor sparse matrices the computation time needed for theadjacent (or perturbed) operators exp(119862

119896119905119896) of the scheme is

much less than that of the main operators exp(119860119905) exp(119861119905)and so the scheme is very effective and can be used as aninitial solution of the iterative schemes

Remark 4 We can also generalize the application of theZassenhaus formula to decompositionmethods with existingBCH formulas for example

(i) 119860-119861 splitting or Lie-Trotter splitting(ii) Strang splitting

Example 5 (1)A-B or Lie-Trotter Splitting For the Lie-Trottersplitting there exists a BCH formula andwe have derived thepolynomials in Theorem 3

(2) Strang SplittingThere exists also a BCH formula which isgiven as

exp (119860119905

2) exp (119861119905) exp (

119860119905

2)= exp (119905119878

1+ 11990531198783+ 11990551198785+sdot sdot sdot )

(16)

where the 119878119894(119860 119861) isin L(119860 119861) is a homogeneous Lie

polynomial in 119860 and 119861 of degree 119894 see [21]The BCH formula is given as

exp ((119860

2+

119861

2) 119905) = Π

infin

119896=2exp (119862

119896119905119896) exp (

119860

2119905) exp(

119861

2119905)

exp((119861

2+

119860

2) 119905) = exp (

119861

2119905) exp(

119860

2119905)Πinfin

119896=2exp (119862

119896119905119896)

(17)

4 International Journal of Differential Equations

and so there exists a new product based on the Zassenhauspolynomials as follows

Πinfin

119896=3exp (119863

119896119905119896) = Π

infin

119896=2exp (119862

119896119905119896)Πinfin

119896=2exp (119862

119896119905119896) (18)

and we obtain a higher order scheme see also [22]

Remark 6 The application of the Zassenhaus formula topartial differential equations for example second-orderparabolic evolution equations needs additional stringentregularity conditions on the operators Due to the nestedcommutators which are needed to apply the Zassenhausformula one needs at least higher regularity conditions

We need some functional analytical results on thedomains of the operators The domains of the operators haveto satisfy the following compatibility conditions for the firstZassenhaus exponents

(i) Condition for the first term 1198622of the Zassenhaus

formula

D (1198712) sub D (119860

2) capD (119860119861) capD (119861119860) capD (119861

2) (19)

where119871 = 119860+119861 andD is the domain of the operators(ii) Condition for the second term 119862

3of the Zassenhaus

formula

D (1198713) sub D (119860

3) capD (119860

2119861)

capD (119860119861119860) capD (1198601198612) capD (119861

3)

(20)

where119871 = 119860+119861 andD is the domain of the operators

Example 7 If we assume to apply the Zassenhaus formula to asecond-order diffusion equation and we assume to split eachspatial direction see the functional analytical framework[23] where the domains are given as

D (119871) = 1198672(Ω) cap 119867

1

0(Ω)

D (119860) = 119888 isin 1198712(Ω) 120597

119909119909119888 120597119909119888 isin 1198712Ω

119906 (0 119910) = 119906 (1 119910) = 0 forall119910 isin (0 1)

D (119861) = 119888 isin 1198712(Ω) 120597

119910119910119888 120597119910119888 isin 1198712Ω

119906 (119909 0) = 119906 (119909 1) = 0 forall119909 isin (0 1)

(21)

whereΩ = (0 1)2

Then we need the following regularity result

(i) condition for the first term 1198622of the Zassenhaus

formula

D (1198712) sub 119867

4(Ω) (22)

(ii) condition for the second term 1198623of the Zassenhaus

formula

D (1198713) sub 119867

6(Ω) (23)

In the example we need a very high regular domain toapply a first or second Zassenhaus exponent This is one ofthe limitations and drawbacks of applying the Zassenhausformula In our applications we restrict on applying the firstor at least the second Zassenhaus exponent We apply thethird and fourth Zassenhaus exponents to the benchmarkproblems given as matrix equations without such restrictionsto the domains see [20]

23 Embedding the Zassenhaus Expansion into an IterativeSplitting Scheme We now discuss the embedding of theZassenhaus formula into the iterative operator splittingschemes

Definition 8 We apply the Lie-Trotter splitting with theembedded Zassenhaus formula which is given as

119864ZassenComp1 (119905) = exp (119860119905) exp (119861119905)

119864ZassenComp119895 (119905) = exp (119862119895119905119895) for 119895 isin 2 119894

(24)

where the sequential Zassenhaus operators are defined as

119864Zassen119894 (119905) = Π119894

119895=1119864ZassenComp119895 (119905) (25)

and we obtain an accuracy of O(119905119894) and 119894 is the number of

Zassenhaus components

In the following theorem we present the improvement ofthe iterative splitting scheme with the Zassenhaus expansion

Theorem 9 One solves the initial value problem (1) Weassume bounded and constant operators 119860 119861

The initialisation process is done with the Zassenhausformula

119888119894(119905) = 119864

119885119886119904119904119890119899119894(119905) 1198880 (26)

where 119864119885119886119904119904119890119899119894

(119905) is given in (25)Furthermore the improved solutions are embedded in the

iterative splitting scheme (4) and one has after 119895 iterative stepsthe following result

119888119894+119895

(119905) = 119864119894119905119890119903119895

(119905) 119864119885119886119904119904119890119899119894

(119905) 1198880 (27)

This has improved the error of the iterative scheme to O(119905119894+119895

)

Proof The solution of the iterative splitting scheme (4) is

119888119894+119895

(119905) = 119864iter119894 (119905) 119888init119895 (28)

where 119878119894(119905) = 119864iter119894(119905) given inTheorem 1

The initialisation is given by the Zassenhaus formula

119888init119895 (119905) = 119864Zassen119895 (119905) 1198880 (29)

Combining both splitting schemes we have a local error of

119890119894+119895 (119905) =

10038171003817100381710038171003817119888 (119905) minus 119888

119894+119895 (119905)10038171003817100381710038171003817

=10038171003817100381710038171003817119888 (119905) minus 119864iter119894 (119905) 119864Zassen119895 (119905) 1198880

10038171003817100381710038171003817

le O (119905119894+119895

) 1198880

(30)

International Journal of Differential Equations 5

Remark 10 We obtain an acceleration of the iterative schemeby applying cheap higher order Zassenhaus formula Such abenefit allows reducing to 2-3 iterative steps while the initialsolution is of a higher order Taking into account the sparsityor nilpotency of the commutators we nowhave a fast iterativesplitting scheme

Remark 11 For our numerical experiment we apply theZassenhaus formulaat the beginning of each integration stepas the so-called preprocessor Therefore we obtain moreaccurate starting conditions for the integration steps of theiterative splitting scheme Based on the regularity conditionsof the nested commutator we apply only 1-2 Zassenhausexponents

In the following we explain the final algorithm ofthe embedded Zassenhaus formula to the iterative splittingscheme

Algorithm 12 We compute 119873 time steps where each timestep Δ119905

119899is given as Δ119905

119899= 119905119899+1

minus 119905119899 with 119899 = 0 119873 minus 1

Further 119888(1199050) is the initial condition of the Cauchy problem

(1) and we compute 119888(1199051) 119888(119905119873) successively We have thefollowing two steps first we apply the Zassenhaus formulaas initialization of 119888

0(119905119899+1

) for the iterative scheme secondwe apply the iterative splitting scheme to the initial valueproblem (1) and compute the final solution 119888(119905

119899+1) as follows

(1) We start with 119895 = 2 and compute the Zassenhausformula

(2) We compute the matrix norm for the Zassenhaussexponents 119862

119895

If errZass119895

le err1 then we start with the iterative

splitting scheme with 1198880(119905) = 119888Zass

119895

119894 = 1 and go to(3) else 119895 = 119895 + 1 and we go to (1)

(3) If the error of the iterative scheme erriter le err2 then

we are finished else 119894 = 119894+1 and we compute the nextiterative step till erriter le err

2

3 Numerical Examples

In this section we discuss examples of the use of embeddingZassenhaus product methods in iterative splitting schemesIn the first example we demonstrate somewhat artificiallyhow the proposed Zassenhaus splitting method avoids thesplitting error up to a certain order The next example showsthe benefit of the combined splitting schemes for applicationsto multiphysics problems

In the following we consider a benchmark example tovalidate the theoretical results and real life problems to applyour schemes

We applied the following splitting schemes given inTable 1

31 Benchmark Problem Matrix Examples In the followingthe idea of thematrix example is to present the decompositioninto two simpler matrices In context of the multiphysics

Table 1 Applied splitting schemes

Abbreviation inthe examples Splitting methods

A-B A-B splittingStrang Strang splitting1198881

Iterative splitting with one iterative step

1198882

Iterative splitting with one iterative step and oneZassenhaus step as prestep

1198883

Iterative splitting with one iterative step and twoZassenhaus steps as presteps

1198884

Iterative splitting with two iterative steps andtwo Zassenhaus steps as presteps

1198885

Iterative splitting with three iterative steps andtwo Zassenhaus steps as presteps

1198886

Iterative splitting with four iterative steps andtwo Zassenhaus steps as presteps

problems even such simple problems can be seen as testexamples We assume that each simple matrix is presenting abasic single physics problems for example reaction processbetween two species while the combination of the twomatrices results in a more complicate physics problem forexample coupled reaction process between two species see[2]

We consider the matrix equation

1199061015840(119905) = [

1 2

3 0] 119906 119906 (0) = 119906

0= (

0

1) 119905 isin [0 1] (31)

the exact solution of which is

119906 (119905) =2 (1198903119905

minus 119890minus2119905

)

5 (32)

We split the matrix as

[1 2

3 0] = [

1 1

1 0] + [

0 1

2 0] = 119860 + 119861 (33)

where [119860 119861] = 0 For this simple example we did not obtainnilpotent matrices for the commutators

We apply 10ndash100 time steps for the computations Thefinal integration time of each single time step was less thanlt1 [sec] such that we need at least for such benchmarkproblems about 1ndash10 [sec] Figure 1 presents the numericalerrors that is the difference between the exact and thenumerical solution

Figure 2 presents the CPU times needed by the standardand the iterative splitting schemes The application of theZassenhaus formula did not increase while the next Zassen-haus coefficients are at least nearly nilpotent matrices and wecan neglect the computational timeMoreover different time-steps did not influence the computations of the Zassenhausexponents which is a preprocess First we compute and saveall the Zassenhaus exponents and later we multiply with thenecessary time steps

6 International Journal of Differential Equations

102

102

10minus2

10minus4

10minus6

10minus8

10minus2

10minus1

100

AB Strang one-sided

AB

Δt

Strang

c1

c2

c3

c4

c5

c6

Erro

r L1

(a)

AB

Δt

Strang

102

102

10minus2

10minus4

10minus6

10minus8

10minus2

10minus1

100

AB Strang two-sided

c1

c2

c3

c4

c5

c6

Erro

r L1

(b)

Figure 1 Numerical errors of the standard splitting scheme and theiterative schemes with the Zassenhaus formula where we have atleast 1 4 iterative steps

Remark 13 We see that the errors decrease significantlywith increasing order of approximation for the initialisa-tion process from the Zassenhaus splitting (we apply 1ndash3Zassenhaus coefficients) Here we apply commutators whichdid not reduce their matrix rank therefore we have thesame computation time as for the pure iterative schemesOverall we have their benefits in the application of theZassenhaus product schemes to the standard Lie-Trotter orStrang-Marchuk splitting Similar results are given withmoreiterative steps 119894 = 3 6 as expected

32 Real Life Problems Multiphase Examples In the nextexamples we simulate coupled transport and reaction equa-tions with different phases so-called multiphase problems

CPU

tim

e

AB Strang one-sided

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

(a)

CPU

tim

e

AB Strang two-sided

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

(b)

Figure 2 CPU times of the standard splitting scheme and theiterative schemes with the Zassenhaus formula where we have atleast 1 4 iterative steps

see [9]The equations are coupledwith the reaction terms andare presented as follows

120597119905119877119894119906119894+ nabla sdot v119906

119894= minus 120582

119894119877119894119906119894+ 120582119894minus1

119877119894minus1

119906119894minus1

+ 120573 (minus119906119894+ 119892119894) inΩ times (0 119879)

1199061198940

(119909) = 119906119894(119909 0) on Ω

120597119905119877119894119892119894= minus 120582

119894119877119894119892119894+ 120582119894minus1

119877119894minus1

119892119894minus1

+ 120573 (minus119892119894+ 119906119894) in Ω times (0 119879)

1198921198940

(119909) = 119892119894(119909 0) on Ω

119894 = 1 119898

(34)

International Journal of Differential Equations 7

where 119898 is the number of equations and 119894 is the index ofeach component The unknown mobile concentrations 119906

119894=

119906119894(119909 119905) are taken in Ω times (0 119879) sub R119899 times R+ where 119899 is the

spatial dimension The unknown immobile concentrations119892119894= 119892119894(119909 119905) are taken in Ω times (0 119879) sub R119899 times R+ where 119899 is

the spatial dimensionThe retardation factors119877119894are constant

and 119877119894

ge 0 The kinetic part is given by the factors 120582119894

They are constant and 120582119894ge 0 For the initialisation of the

kinetic part we set 1205820

= 0 The kinetic part is linear andirreversible so the successors have only one predecessor Theinitial conditions are given for each component 119894 as constantsor linear impulses For the boundary conditions we havetrivial inflow and outflow conditions with 119906

119894= 0 at the inflow

boundary The transport part is given by the velocity v isin R119899

and is piecewise constant see [2] The exchange between themobile and immobile part is given by 120573

In the following we discuss the one-phase and two-phaseexamples

321 One-Phase Example The next example is a simplifiedreal life problem with a multiphase transport-reaction equa-tion We treat the mobile and immobile pores in a porousmedia Such simulations are made about waste scenarios see[2 24]

We concentrate on the computational benefits of a fastcomputation of the mixed iterative scheme with the Zassen-haus formula

The one phase equation is

1205971199051198881+ nabla sdot F

11198881= minus12058211198881 in Ω times [0 119905]

1205971199051198882+ nabla sdot F

21198882= 12058211198881minus 12058221198882 in Ω times [0 119905]

F119894= v119894minus 119863119894nabla 119894 = 1 2

1198881(x 119905) = 119888

10(x) 119888

2(x 119905) = 119888

20(x) on Ω

1198881 (x 119905) = 119888

11 (x 119905) 1198882 (x 119905) = 119888

21 (x 119905)

on 120597Ω times [0 119905]

(35)

Here the parameters take the values V1= 01 V

2= 005119863

1=

0011198632= 0005 and 120582

1= 1205822= 01

We now turn to the finite difference schemes andsemidiscretise the equation with its convection and diffusionoperators as follows

120597119905c = (119860

1+ 1198602) c (36)

We obtain the two matrices and decouple the diffusion andconvection parts as follows

1198601= (

119860diff 0

0 119860diff) isin R

2119868times2119868

1198602= (

119860Conv 0

0 119860Conv) + (

minusΛ1

0

Λ1

minusΛ2

) isin R2119868times2119868

(37)

We apply the splitting method to the operators 1198601and 119860

2

given in Section 21

The submatrices are

119860diff119894 =119863119894

Δ11990921198771=

119863119894

Δ1199092sdot (

minus2 1

1 minus2 1

d d d1 minus2 1

1 minus2

) isin R119868times119868

119860conv119894 = minusV119894

Δ1199091198772= minus

V119894

Δ119909sdot (

1

minus1 1

d dminus1 1

minus1 1

) isin R119868times119868

(38)

where 119868 is the number of spatial points and Δ119909 is the spatialstep size Consider

Λ1= (

1205821

0

0 1205821

0

d d d0 1205821

0

0 1205821

) isin R119868times119868

Λ2= (

1205822

0

0 1205822

0

d d d0 1205822

0

0 1205822

) isin R119868times119868

(39)

We have the commutator

[1198601 1198602] = (

0 0

1198632minus 1198631

Δ11990921198771Λ1

0) isin R

2119868times2119868 (40)

where we assume that [1198771 1198772] asymp 0 This is a nilpotent

commutator and the computation time needed for thecommutators of such operators is less than that for the fulloperators Furthermore we assume that we have a boundedspatial step size which is given by Δ119909 = 01 ≫ 0 so thatwe scale the singular entry (119863

2minus 1198631)Δ1199092asymp 1 and evade a

blowup in such matricesWe have obtained the optimal results of a basically

constant CPU time for decreasing time steps which increasein the standard scheme

We apply 10ndash100 timesteps for the computations with 10ndash100 spatial points The final integration time of each singletime step was about 1ndash10 [sec] such that we need at least forsuch benchmark problems about 100ndash1000 [sec]

Figure 3 presents the numerical error that is the differ-ence between the exact and the numerical solution

Figure 4 presents the CPU time of the standard andthe iterative splitting schemes Based on the preprocess ofcomputing the Zassenhaus exponents at the beginning andstore the matrices we could save additional computationaltime with larger time steps

Remark 14 For the iterative schemes with embedded Zassen-haus products we can reach improved results more quicklyThe benefits come from the constant CPU time needed for

8 International Journal of Differential Equations

AB

Δt

Strang

101

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

Erro

r L1

AB Strang one-sided with A

(a)

AB

Δt

Strang

102

100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

Erro

r L1

AB Strang one-sided with B

(b)

AB

Δt

Strang

102

100

10minus2

10minus4

10minus6

10minus8

10minus2

10minus1

100

AB Strang two-side

c1

c2

c3

c4

c5

c6

Erro

r L1

(c)

Figure 3 Numerical errors of the standard splitting scheme and theiterative schemes with the Zassenhaus formula where we have atleast 1 4 iterative steps

CPU

tim

e

AB strang one-side with A

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

(a)

CPU

tim

e

Δt

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

AB

Strang

c1

c2

c3

c4

c5

c6

AB Strang one-sided with B

(b)

CPU

tim

e

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

AB Strang two-side

c1

c2

c3

c4

c5

c6

(c)

Figure 4 CPU time of the standard splitting scheme and theiterative schemes with the Zassenhaus formula where we have atleast 1 4 iterative steps

International Journal of Differential Equations 9

calculating the nilpotent commutators in the Zassenhausformula So it makes sense to have 1ndash3 Zassenhaus stepsbefore starting the iterative scheme At that point only 2-3iterative steps are needed to obtainmore accurate results thanthose from the expensive standard schemes Here too weobtain the best results with the one-sided iterative schemes

Remark 15 Based on our restriction to 1198631 1198632 and Δ119909 with

(1198632minus 1198631)Δ1199092

asymp 1 we can control the delicate blowupof the computed commutators By the way if we neglectthe restriction and Δ119909 with (119863

2minus 1198631)Δ1199092

rarr infin thecommutator blows up and we lose the benefits of such analgorithmThis is a limitation of applying such an embeddedZassenhaus formula

322 Two Phase Example The next example is a moredelicate real life problem involving a multiphase transport-reaction equation We treat mobile and immobile pores in aporousmedia Such simulations aremade forwaste scenariossee [2]

We concentrate on the computational benefits of a fastcomputation of the commutators and their use for theinitialisation of the iterative splitting scheme

The equation is

1205971199051198881+ nabla sdot F119888

1= 119892 (minus119888

1+ 1198881119894119898

) minus 12058211198881 in Ω times [0 119905]

1205971199051198882+ nabla sdot F119888

2= 119892 (minus119888

2+ 1198882119894119898

) + 12058211198881minus 12058221198882 in Ω times [0 119905]

F = v minus 119863nabla

1205971199051198881119894119898

= 119892 (1198881minus 1198881119894119898

) minus 12058211198881119894119898

in Ω times [0 119905]

1205971199051198882119894119898

= 119892 (1198882minus 1198882119894119898

) + 12058211198881119894119898

minus 12058221198882119894119898

in Ω times [0 119905]

1198881(x 119905) = 119888

10(x) 119888

2(x 119905) = 119888

20(x) on Ω

1198881(x 119905) = 119888

11(x 119905) 119888

2(x 119905) = 119888

21(x 119905)

on 120597Ω times [0 119905]

1198881119894119898

(x 119905) = 0 1198882119894119898

(x 119905) = 0 on Ω

1198881119894119898 (x 119905) = 0 119888

2119894119898 (x 119905) = 0 on 120597Ω times [0 119905]

(41)

Here the parameters are V = 01 119863 = 001 1205821= 1205822= 01

and 119892 = 001We next treat the semidiscretised equation given by the

matrices

120597119905C = (

119860 minus Λ1minus 119866 0 119866 0

Λ1

119860 minus Λ2minus 119866 0 119866

119866 0 minusΛ1minus 119866 0

0 119866 Λ1

minusΛ2minus 119866

)C

(42)

where C = (c1 c2 c1119894119898 c2119894119898)119879 while c1 = (119888

11 119888

1119868) is the

solution of the first species in themobile phase in each spatialdiscretisation point (119894 = 1 119868) and the same for the othersolution vectors

We have the following two operators for the splittingmethod

119860 =119863

Δ1199092sdot (

minus2 1

1 minus2 1

d d d1 minus2 1

1 minus2

)

+VΔ119909

sdot (

1

minus1 1

d dminus1 1

minus1 1

) isin R119868times119868

(43)

where 119868 is the number of spatial pointsConsider

Λ1= (

1205821

0

0 1205821

0

d d d0 1205821

0

0 1205821

)

Λ2= (

1205822

0

0 1205822

0

d d d0 1205822

0

0 1205822

) isin R119868times119868

119866 = (

119892 0

0 119892 0

d d d0 119892 0

0 119892

) isin R119868times119868

(44)

We decouple this into the following matrices

119860 = (

119860 0 0 0

0 119860 0 0

0 0 0 0

0 0 0 0

) isin R4119868times4119868

1198611= (

minusΛ1

0 0 0

Λ1

minusΛ2

0 0

0 0 minusΛ1

0

0 0 Λ1

minusΛ2

)

1198612= (

minus119866 0 119866 0

0 minus119866 0 119866

119866 0 minus119866 0

0 119866 0 minus119866

) isin R4119868times4119868

(45)

For the operators 119860 and 119861 = 1198611+ 1198612and 119861 we apply the

iterative splitting method given in Section 21Based on the decomposition119860 is block diagonal and 119861 is

tridiagonalThe commutator is

[119860 119861] = (

0 0 119860119866 0

0 0 0 119860119866

minus119860119866 0 0 0

0 minus119860119866 0 0

) isin R4119868times4119868

(46)

10 International Journal of Differential Equations

Strang

AB

102

101

100

10minus1

10minus2

10minus3

100

Δt

c1

c2

c3

c4

c5

c6

Erro

r L1

AB Strang one-sided with B

(a)

CPU

tim

e

AB Strang one-side

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

(b)

Figure 5 Numerical errors and CPU time of the standard splittingscheme and the iterative schemes with Zassenhaus formula wherewe have at least 1 4 iterative steps

where we have a sparser operator than the main operator 119861and therefore we save computation time in computing suchoperators Further we assume that we have a bounded spatialstep size which is given by Δ119909 = 01 ≫ 0 so that we scale thesingular entry119863Δ119909

2asymp 1 and VΔ119909 asymp 1

Figure 5 presents the numerical error that is the differ-ence between the exact and the numerical solution and theCPU time required The optimal results are from the one-sided iterative schemes on the operator 119861

Remark 16 We obtain the same results as in the one-phase example because of the basically constant CPUtime for decreasing time steps of the iterative scheme weobtain optimal results when applying 1ndash3 Zassenhaus stepsTherefore with the embedded Zassenhaus formula we can

reach improved results more quickly than with the standardschemes With 2-3 more iterative steps we obtain moreaccurate results With the one-sided iterative schemes wereach the best convergence results when using the moredelicate operator 119861

4 Conclusions and Discussion

In this paper we have presented a novel splitting schemecombining the ideas of iterative and sequential schemes basedon the Zassenhaus formula Here the idea is to decouplethe expensive iterative computation of the schemes basedonly on the matrix exponential obtaining simpler embeddedZassenhaus schemes based on nilpotent commutators Suchcombinations have the benefit of reducing the computationtime because the commutators can be computed very cheaplyOn the other hand simple linear iterative steps can be donevery cheaply if they are initialised with a sufficiently accuratestarting solution and this accelerates the solvers The erroranalysis showed that these are stablemethods for higher orderschemes In the applications we were able to demonstratethe speedup from the Zassenhaus enhanced method andwere able to apply it to multiphysics applications In thefuture we will concentrate on linear and nonlinear matrix-dependent schemes that switch between noniterative anditerative schemes based on Zassenhaus products

References

[1] R E Ewing ldquoUp-scaling of biological processes andmultiphaseow in porous mediardquo in IIMA Volumes in Mathematics andIts Applications vol 295 pp 195ndash215 Springer New York NYUSA 2002

[2] J Geiser Discretization and Simulation of Systems forConvection-Diffusion-Dispersion Reactions with Applicationsin Groundwater Contamination Groundwater ModellingManagement and Contamination Nova Science MonographNew York NY USA 2008

[3] N Antonic C J van Duijn W Jager and A MikelicMultiscaleProblems in Science and Technology Challenges toMathematicalAnalysis and Perspectives Springer Berlin Germany 2002

[4] E Weinan Principle of Multiscale Modelling Cambridge Uni-versity Press Cambridge UK 2011

[5] S Blanes and F Casas ldquoOn the necessity of negative coefficientsfor operator splitting schemes of order higher than twordquoAppliedNumerical Mathematics vol 54 no 1 pp 23ndash37 2005

[6] EHansen andAOstermann ldquoHigh order splittingmethods foranalytic semigroups existrdquo BIT NumericalMathematics vol 49no 3 pp 527ndash542 2009

[7] G Strang ldquoOn the construction and comparison of differenceschemesrdquo SIAM Journal on Numerical Analysis vol 5 pp 506ndash517 1968

[8] G I Marchuk ldquoSome applicatons of splitting-up methods tothe solution of problems in mathematical physicsrdquo AplikaceMatematiky vol 1 pp 103ndash132 1968

[9] J Geiser ldquoComputing exponential for iterative splitting meth-ods algorithms and applicationsrdquo Journal of AppliedMathemat-ics vol 2011 Article ID 193781 27 pages 2011

[10] R I McLachlan and G RW Quispel ldquoSplittingmethodsrdquoActaNumerica vol 11 pp 341ndash434 2002

International Journal of Differential Equations 11

[11] J Geiser G Tanoglu and N Gucuyenen ldquoHigher order oper-ator splitting methods via Zassenhaus product formula theoryand applicationsrdquo Computers amp Mathematics with Applicationsvol 62 no 4 pp 1994ndash2015 2011

[12] G H Weiss and A A Maradudin ldquoThe Baker-Hausdorff for-mula and a problem in crystal physicsrdquo Journal of MathematicalPhysics vol 3 pp 771ndash777 1962

[13] R MWilcox ldquoExponential operators and parameter differenti-ation in quantum physicsrdquo Journal of Mathematical Physics vol8 pp 962ndash982 1967

[14] J Geiser ldquoAn iterative splitting approach for linear integro-differential equationsrdquo Applied Mathematics Letters vol 26 no11 pp 1048ndash1052 2013

[15] F Casas and A Murua ldquoAn efficient algorithm for computingthe Baker-Campbell-Hausdorff series and some of its applica-tionsrdquo Journal of Mathematical Physics vol 50 no 3 Article ID033513 2009

[16] J Geiser Iterative Splitting Methods for Differential EquationsChapmanampHallCRCNumerical Analysis and Scientific Com-puting CRC Press Boca Raton Fla USA 2011

[17] S Vandewalle Parallel Multigrid Waveform Relaxation forParabolic Problems Teubner Stuttgart Germany 1993

[18] J Geiser and G Tanoglu ldquoOperator-splitting methods viathe Zassenhaus product formulardquo Applied Mathematics andComputation vol 217 no 9 pp 4557ndash4575 2011

[19] F Bayen ldquoOn the convergence of the Zassenhaus formulardquoLetters in Mathematical Physics vol 3 no 3 pp 161ndash167 1979

[20] F Casas A Murua and M Nadinic ldquoEfficient computation ofthe Zassenhaus formulardquo Computer Physics Communicationsvol 183 no 11 pp 2386ndash2391 2012

[21] E Hairer C Lubich and GWannerGeometric Numerical Inte-gration vol 31 of Springer Series in Computational MathematicsSpringer Berlin Germany 2002

[22] H Yoshida ldquoConstruction of higher order symplectic integra-torsrdquo Physics Letters A vol 150 no 5ndash7 pp 262ndash268 1990

[23] E Faou A Ostermann and K Schratz ldquoAnalysis of exponentialsplitting methods for inhomogeneous parabolic equationsrdquohttparxivorgabs12125827

[24] J Geiser ldquoDiscretization methods with embedded analyticalsolutions for convection-diffusion dispersion-reaction equa-tions and applicationsrdquo Journal of EngineeringMathematics vol57 no 1 pp 79ndash98 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Differential Equations 3

+ sdot sdot sdot + int

119905119899+1

119905119899

exp (119860 (119905119899+1

minus 1199041)) 119861

sdot sdot sdot int

119905119899+1minussum119894minus1

119895=1119904119895

119905119899

exp((119905119899+1

minus

119894

sum

119895=1

119904119895)119860)

times 119861 exp ((119904119894(119860 + 119861)) 119888 (119905

119899) 119889119904119894 119889119904

21198891199041

(7)

We obtain1003817100381710038171003817119890119894

1003817100381710038171003817 le1003817100381710038171003817exp ((119860 + 119861) 120591) 119888 (119905

119899) minus 119878119894 (120591) 119888init (120591)

1003817100381710038171003817

le 119862120591119894 max119904119894isin[0120591]

1003817100381710038171003817exp (119904119894(119860 + 119861)) 119888 (119905

119899) minus 119888init (119904119894)

1003817100381710038171003817

le 119862120591119894 1003817100381710038171003817exp (120591 (119860 + 119861)) 119888 (119905

119899) minus 119888init (120591)

1003817100381710038171003817

(8)

where 119894 is the number of iterative stepsThe same idea can be applied to the even iterative scheme

and also for alternating 119860 and 119861

Remark 2 The accuracy of the initialisation 119888init(120591) is impor-tant for conserving or improving the underlying iterativesplitting scheme

Here we have the following initialisation schemes

1198881(120591) = exp (119860120591) 119888 (119905

119899) 997888rarr

10038171003817100381710038171198901198941003817100381710038171003817 le 119862120591

119894119888 (119905119899)

1198881(120591) = exp (119860120591) exp (119861120591) 119888 (119905

119899) 997888rarr

10038171003817100381710038171198901198941003817100381710038171003817 le 119862120591

119894+1119888 (119905119899)

(9)

So the slow convergence in the initial process 1198881(120591) of the

iterative splitting schemes can be improved by a cheapcomputable initial process 119888

1(120591) asymp exp((119860 + 119861)120591) We will

next discuss the fast convergent Zassenhaus formula for theinitial process

22 Zassenhaus Formula (Sequential Splitting) The Zassen-haus formula is an extension to the exponential splittingschemes and can be given inTheorem 3 see also [20]

Theorem 3 One assumes L(119860 119861) is the free Lie algebragenerated by 119860 and 119861 The kernel function of solution (1)exp((119860 + 119861)119905) can be uniquely decomposed as

exp ((119860 + 119861) 119905) = exp (119860119905) exp (119861119905) 120587119895

119894=1120587119898

119896=2

times exp (119862119896 (119860 119861) 119905

119896)

+ O (119905119898+1

)

(10)

where 119862119895(119860 119861) isin L(119860 119861) is a homogeneous Lie polynomial

of 119860 and 119861 of degree 119895

Proof The sketch of the proof is given in [20] The ideais based on the existence of the BCH (Baker-Campbell-Hausdorff) formula and we obtain

exp (minus119860119905) exp ((119860 + 119861) 119905) = exp (119861119905 + 119863 (119905)) (11)

where119863(119905) is a Lie polynomial of degree gt1 and

exp ((119860 + 119861) 119905) = exp (119860119905) exp (119861119905) + O (1199052) (12)

Further we have

exp (minus119861119905) exp (119861119905 + 119863 (119905)) = exp (11986221199052+ 119863 (119905)) (13)

where119863(119905) is a Lie polynomial of degree gt2 and

exp ((119860 + 119861) 119905) = exp (119860119905) exp (119861119905) exp (11986221199052) + O (119905

3)

(14)

We can repeat the recursive process and obtain by completeinduction the results

The first Zassenhaus polynomials are given as

1198622(119860 119861) =

1

2[119861 119860]

1198623 (119860 119861) =

1

3[[119861 119860] 119861] +

1

6[[119861 119860] 119860]

1198624 (119860 119861) =

1

24[[[119861 119860] 119860] 119860] +

1

8[[[119861 119860] 119861] 119860]

+1

8[[[119861 119860] 119861] 119861]

(15)

Here we see the benefit of such schemes in the computationof the commutators If we assume that we have quicklycomputable commutators for example nilpotent matricesor sparse matrices the computation time needed for theadjacent (or perturbed) operators exp(119862

119896119905119896) of the scheme is

much less than that of the main operators exp(119860119905) exp(119861119905)and so the scheme is very effective and can be used as aninitial solution of the iterative schemes

Remark 4 We can also generalize the application of theZassenhaus formula to decompositionmethods with existingBCH formulas for example

(i) 119860-119861 splitting or Lie-Trotter splitting(ii) Strang splitting

Example 5 (1)A-B or Lie-Trotter Splitting For the Lie-Trottersplitting there exists a BCH formula andwe have derived thepolynomials in Theorem 3

(2) Strang SplittingThere exists also a BCH formula which isgiven as

exp (119860119905

2) exp (119861119905) exp (

119860119905

2)= exp (119905119878

1+ 11990531198783+ 11990551198785+sdot sdot sdot )

(16)

where the 119878119894(119860 119861) isin L(119860 119861) is a homogeneous Lie

polynomial in 119860 and 119861 of degree 119894 see [21]The BCH formula is given as

exp ((119860

2+

119861

2) 119905) = Π

infin

119896=2exp (119862

119896119905119896) exp (

119860

2119905) exp(

119861

2119905)

exp((119861

2+

119860

2) 119905) = exp (

119861

2119905) exp(

119860

2119905)Πinfin

119896=2exp (119862

119896119905119896)

(17)

4 International Journal of Differential Equations

and so there exists a new product based on the Zassenhauspolynomials as follows

Πinfin

119896=3exp (119863

119896119905119896) = Π

infin

119896=2exp (119862

119896119905119896)Πinfin

119896=2exp (119862

119896119905119896) (18)

and we obtain a higher order scheme see also [22]

Remark 6 The application of the Zassenhaus formula topartial differential equations for example second-orderparabolic evolution equations needs additional stringentregularity conditions on the operators Due to the nestedcommutators which are needed to apply the Zassenhausformula one needs at least higher regularity conditions

We need some functional analytical results on thedomains of the operators The domains of the operators haveto satisfy the following compatibility conditions for the firstZassenhaus exponents

(i) Condition for the first term 1198622of the Zassenhaus

formula

D (1198712) sub D (119860

2) capD (119860119861) capD (119861119860) capD (119861

2) (19)

where119871 = 119860+119861 andD is the domain of the operators(ii) Condition for the second term 119862

3of the Zassenhaus

formula

D (1198713) sub D (119860

3) capD (119860

2119861)

capD (119860119861119860) capD (1198601198612) capD (119861

3)

(20)

where119871 = 119860+119861 andD is the domain of the operators

Example 7 If we assume to apply the Zassenhaus formula to asecond-order diffusion equation and we assume to split eachspatial direction see the functional analytical framework[23] where the domains are given as

D (119871) = 1198672(Ω) cap 119867

1

0(Ω)

D (119860) = 119888 isin 1198712(Ω) 120597

119909119909119888 120597119909119888 isin 1198712Ω

119906 (0 119910) = 119906 (1 119910) = 0 forall119910 isin (0 1)

D (119861) = 119888 isin 1198712(Ω) 120597

119910119910119888 120597119910119888 isin 1198712Ω

119906 (119909 0) = 119906 (119909 1) = 0 forall119909 isin (0 1)

(21)

whereΩ = (0 1)2

Then we need the following regularity result

(i) condition for the first term 1198622of the Zassenhaus

formula

D (1198712) sub 119867

4(Ω) (22)

(ii) condition for the second term 1198623of the Zassenhaus

formula

D (1198713) sub 119867

6(Ω) (23)

In the example we need a very high regular domain toapply a first or second Zassenhaus exponent This is one ofthe limitations and drawbacks of applying the Zassenhausformula In our applications we restrict on applying the firstor at least the second Zassenhaus exponent We apply thethird and fourth Zassenhaus exponents to the benchmarkproblems given as matrix equations without such restrictionsto the domains see [20]

23 Embedding the Zassenhaus Expansion into an IterativeSplitting Scheme We now discuss the embedding of theZassenhaus formula into the iterative operator splittingschemes

Definition 8 We apply the Lie-Trotter splitting with theembedded Zassenhaus formula which is given as

119864ZassenComp1 (119905) = exp (119860119905) exp (119861119905)

119864ZassenComp119895 (119905) = exp (119862119895119905119895) for 119895 isin 2 119894

(24)

where the sequential Zassenhaus operators are defined as

119864Zassen119894 (119905) = Π119894

119895=1119864ZassenComp119895 (119905) (25)

and we obtain an accuracy of O(119905119894) and 119894 is the number of

Zassenhaus components

In the following theorem we present the improvement ofthe iterative splitting scheme with the Zassenhaus expansion

Theorem 9 One solves the initial value problem (1) Weassume bounded and constant operators 119860 119861

The initialisation process is done with the Zassenhausformula

119888119894(119905) = 119864

119885119886119904119904119890119899119894(119905) 1198880 (26)

where 119864119885119886119904119904119890119899119894

(119905) is given in (25)Furthermore the improved solutions are embedded in the

iterative splitting scheme (4) and one has after 119895 iterative stepsthe following result

119888119894+119895

(119905) = 119864119894119905119890119903119895

(119905) 119864119885119886119904119904119890119899119894

(119905) 1198880 (27)

This has improved the error of the iterative scheme to O(119905119894+119895

)

Proof The solution of the iterative splitting scheme (4) is

119888119894+119895

(119905) = 119864iter119894 (119905) 119888init119895 (28)

where 119878119894(119905) = 119864iter119894(119905) given inTheorem 1

The initialisation is given by the Zassenhaus formula

119888init119895 (119905) = 119864Zassen119895 (119905) 1198880 (29)

Combining both splitting schemes we have a local error of

119890119894+119895 (119905) =

10038171003817100381710038171003817119888 (119905) minus 119888

119894+119895 (119905)10038171003817100381710038171003817

=10038171003817100381710038171003817119888 (119905) minus 119864iter119894 (119905) 119864Zassen119895 (119905) 1198880

10038171003817100381710038171003817

le O (119905119894+119895

) 1198880

(30)

International Journal of Differential Equations 5

Remark 10 We obtain an acceleration of the iterative schemeby applying cheap higher order Zassenhaus formula Such abenefit allows reducing to 2-3 iterative steps while the initialsolution is of a higher order Taking into account the sparsityor nilpotency of the commutators we nowhave a fast iterativesplitting scheme

Remark 11 For our numerical experiment we apply theZassenhaus formulaat the beginning of each integration stepas the so-called preprocessor Therefore we obtain moreaccurate starting conditions for the integration steps of theiterative splitting scheme Based on the regularity conditionsof the nested commutator we apply only 1-2 Zassenhausexponents

In the following we explain the final algorithm ofthe embedded Zassenhaus formula to the iterative splittingscheme

Algorithm 12 We compute 119873 time steps where each timestep Δ119905

119899is given as Δ119905

119899= 119905119899+1

minus 119905119899 with 119899 = 0 119873 minus 1

Further 119888(1199050) is the initial condition of the Cauchy problem

(1) and we compute 119888(1199051) 119888(119905119873) successively We have thefollowing two steps first we apply the Zassenhaus formulaas initialization of 119888

0(119905119899+1

) for the iterative scheme secondwe apply the iterative splitting scheme to the initial valueproblem (1) and compute the final solution 119888(119905

119899+1) as follows

(1) We start with 119895 = 2 and compute the Zassenhausformula

(2) We compute the matrix norm for the Zassenhaussexponents 119862

119895

If errZass119895

le err1 then we start with the iterative

splitting scheme with 1198880(119905) = 119888Zass

119895

119894 = 1 and go to(3) else 119895 = 119895 + 1 and we go to (1)

(3) If the error of the iterative scheme erriter le err2 then

we are finished else 119894 = 119894+1 and we compute the nextiterative step till erriter le err

2

3 Numerical Examples

In this section we discuss examples of the use of embeddingZassenhaus product methods in iterative splitting schemesIn the first example we demonstrate somewhat artificiallyhow the proposed Zassenhaus splitting method avoids thesplitting error up to a certain order The next example showsthe benefit of the combined splitting schemes for applicationsto multiphysics problems

In the following we consider a benchmark example tovalidate the theoretical results and real life problems to applyour schemes

We applied the following splitting schemes given inTable 1

31 Benchmark Problem Matrix Examples In the followingthe idea of thematrix example is to present the decompositioninto two simpler matrices In context of the multiphysics

Table 1 Applied splitting schemes

Abbreviation inthe examples Splitting methods

A-B A-B splittingStrang Strang splitting1198881

Iterative splitting with one iterative step

1198882

Iterative splitting with one iterative step and oneZassenhaus step as prestep

1198883

Iterative splitting with one iterative step and twoZassenhaus steps as presteps

1198884

Iterative splitting with two iterative steps andtwo Zassenhaus steps as presteps

1198885

Iterative splitting with three iterative steps andtwo Zassenhaus steps as presteps

1198886

Iterative splitting with four iterative steps andtwo Zassenhaus steps as presteps

problems even such simple problems can be seen as testexamples We assume that each simple matrix is presenting abasic single physics problems for example reaction processbetween two species while the combination of the twomatrices results in a more complicate physics problem forexample coupled reaction process between two species see[2]

We consider the matrix equation

1199061015840(119905) = [

1 2

3 0] 119906 119906 (0) = 119906

0= (

0

1) 119905 isin [0 1] (31)

the exact solution of which is

119906 (119905) =2 (1198903119905

minus 119890minus2119905

)

5 (32)

We split the matrix as

[1 2

3 0] = [

1 1

1 0] + [

0 1

2 0] = 119860 + 119861 (33)

where [119860 119861] = 0 For this simple example we did not obtainnilpotent matrices for the commutators

We apply 10ndash100 time steps for the computations Thefinal integration time of each single time step was less thanlt1 [sec] such that we need at least for such benchmarkproblems about 1ndash10 [sec] Figure 1 presents the numericalerrors that is the difference between the exact and thenumerical solution

Figure 2 presents the CPU times needed by the standardand the iterative splitting schemes The application of theZassenhaus formula did not increase while the next Zassen-haus coefficients are at least nearly nilpotent matrices and wecan neglect the computational timeMoreover different time-steps did not influence the computations of the Zassenhausexponents which is a preprocess First we compute and saveall the Zassenhaus exponents and later we multiply with thenecessary time steps

6 International Journal of Differential Equations

102

102

10minus2

10minus4

10minus6

10minus8

10minus2

10minus1

100

AB Strang one-sided

AB

Δt

Strang

c1

c2

c3

c4

c5

c6

Erro

r L1

(a)

AB

Δt

Strang

102

102

10minus2

10minus4

10minus6

10minus8

10minus2

10minus1

100

AB Strang two-sided

c1

c2

c3

c4

c5

c6

Erro

r L1

(b)

Figure 1 Numerical errors of the standard splitting scheme and theiterative schemes with the Zassenhaus formula where we have atleast 1 4 iterative steps

Remark 13 We see that the errors decrease significantlywith increasing order of approximation for the initialisa-tion process from the Zassenhaus splitting (we apply 1ndash3Zassenhaus coefficients) Here we apply commutators whichdid not reduce their matrix rank therefore we have thesame computation time as for the pure iterative schemesOverall we have their benefits in the application of theZassenhaus product schemes to the standard Lie-Trotter orStrang-Marchuk splitting Similar results are given withmoreiterative steps 119894 = 3 6 as expected

32 Real Life Problems Multiphase Examples In the nextexamples we simulate coupled transport and reaction equa-tions with different phases so-called multiphase problems

CPU

tim

e

AB Strang one-sided

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

(a)

CPU

tim

e

AB Strang two-sided

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

(b)

Figure 2 CPU times of the standard splitting scheme and theiterative schemes with the Zassenhaus formula where we have atleast 1 4 iterative steps

see [9]The equations are coupledwith the reaction terms andare presented as follows

120597119905119877119894119906119894+ nabla sdot v119906

119894= minus 120582

119894119877119894119906119894+ 120582119894minus1

119877119894minus1

119906119894minus1

+ 120573 (minus119906119894+ 119892119894) inΩ times (0 119879)

1199061198940

(119909) = 119906119894(119909 0) on Ω

120597119905119877119894119892119894= minus 120582

119894119877119894119892119894+ 120582119894minus1

119877119894minus1

119892119894minus1

+ 120573 (minus119892119894+ 119906119894) in Ω times (0 119879)

1198921198940

(119909) = 119892119894(119909 0) on Ω

119894 = 1 119898

(34)

International Journal of Differential Equations 7

where 119898 is the number of equations and 119894 is the index ofeach component The unknown mobile concentrations 119906

119894=

119906119894(119909 119905) are taken in Ω times (0 119879) sub R119899 times R+ where 119899 is the

spatial dimension The unknown immobile concentrations119892119894= 119892119894(119909 119905) are taken in Ω times (0 119879) sub R119899 times R+ where 119899 is

the spatial dimensionThe retardation factors119877119894are constant

and 119877119894

ge 0 The kinetic part is given by the factors 120582119894

They are constant and 120582119894ge 0 For the initialisation of the

kinetic part we set 1205820

= 0 The kinetic part is linear andirreversible so the successors have only one predecessor Theinitial conditions are given for each component 119894 as constantsor linear impulses For the boundary conditions we havetrivial inflow and outflow conditions with 119906

119894= 0 at the inflow

boundary The transport part is given by the velocity v isin R119899

and is piecewise constant see [2] The exchange between themobile and immobile part is given by 120573

In the following we discuss the one-phase and two-phaseexamples

321 One-Phase Example The next example is a simplifiedreal life problem with a multiphase transport-reaction equa-tion We treat the mobile and immobile pores in a porousmedia Such simulations are made about waste scenarios see[2 24]

We concentrate on the computational benefits of a fastcomputation of the mixed iterative scheme with the Zassen-haus formula

The one phase equation is

1205971199051198881+ nabla sdot F

11198881= minus12058211198881 in Ω times [0 119905]

1205971199051198882+ nabla sdot F

21198882= 12058211198881minus 12058221198882 in Ω times [0 119905]

F119894= v119894minus 119863119894nabla 119894 = 1 2

1198881(x 119905) = 119888

10(x) 119888

2(x 119905) = 119888

20(x) on Ω

1198881 (x 119905) = 119888

11 (x 119905) 1198882 (x 119905) = 119888

21 (x 119905)

on 120597Ω times [0 119905]

(35)

Here the parameters take the values V1= 01 V

2= 005119863

1=

0011198632= 0005 and 120582

1= 1205822= 01

We now turn to the finite difference schemes andsemidiscretise the equation with its convection and diffusionoperators as follows

120597119905c = (119860

1+ 1198602) c (36)

We obtain the two matrices and decouple the diffusion andconvection parts as follows

1198601= (

119860diff 0

0 119860diff) isin R

2119868times2119868

1198602= (

119860Conv 0

0 119860Conv) + (

minusΛ1

0

Λ1

minusΛ2

) isin R2119868times2119868

(37)

We apply the splitting method to the operators 1198601and 119860

2

given in Section 21

The submatrices are

119860diff119894 =119863119894

Δ11990921198771=

119863119894

Δ1199092sdot (

minus2 1

1 minus2 1

d d d1 minus2 1

1 minus2

) isin R119868times119868

119860conv119894 = minusV119894

Δ1199091198772= minus

V119894

Δ119909sdot (

1

minus1 1

d dminus1 1

minus1 1

) isin R119868times119868

(38)

where 119868 is the number of spatial points and Δ119909 is the spatialstep size Consider

Λ1= (

1205821

0

0 1205821

0

d d d0 1205821

0

0 1205821

) isin R119868times119868

Λ2= (

1205822

0

0 1205822

0

d d d0 1205822

0

0 1205822

) isin R119868times119868

(39)

We have the commutator

[1198601 1198602] = (

0 0

1198632minus 1198631

Δ11990921198771Λ1

0) isin R

2119868times2119868 (40)

where we assume that [1198771 1198772] asymp 0 This is a nilpotent

commutator and the computation time needed for thecommutators of such operators is less than that for the fulloperators Furthermore we assume that we have a boundedspatial step size which is given by Δ119909 = 01 ≫ 0 so thatwe scale the singular entry (119863

2minus 1198631)Δ1199092asymp 1 and evade a

blowup in such matricesWe have obtained the optimal results of a basically

constant CPU time for decreasing time steps which increasein the standard scheme

We apply 10ndash100 timesteps for the computations with 10ndash100 spatial points The final integration time of each singletime step was about 1ndash10 [sec] such that we need at least forsuch benchmark problems about 100ndash1000 [sec]

Figure 3 presents the numerical error that is the differ-ence between the exact and the numerical solution

Figure 4 presents the CPU time of the standard andthe iterative splitting schemes Based on the preprocess ofcomputing the Zassenhaus exponents at the beginning andstore the matrices we could save additional computationaltime with larger time steps

Remark 14 For the iterative schemes with embedded Zassen-haus products we can reach improved results more quicklyThe benefits come from the constant CPU time needed for

8 International Journal of Differential Equations

AB

Δt

Strang

101

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

Erro

r L1

AB Strang one-sided with A

(a)

AB

Δt

Strang

102

100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

Erro

r L1

AB Strang one-sided with B

(b)

AB

Δt

Strang

102

100

10minus2

10minus4

10minus6

10minus8

10minus2

10minus1

100

AB Strang two-side

c1

c2

c3

c4

c5

c6

Erro

r L1

(c)

Figure 3 Numerical errors of the standard splitting scheme and theiterative schemes with the Zassenhaus formula where we have atleast 1 4 iterative steps

CPU

tim

e

AB strang one-side with A

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

(a)

CPU

tim

e

Δt

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

AB

Strang

c1

c2

c3

c4

c5

c6

AB Strang one-sided with B

(b)

CPU

tim

e

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

AB Strang two-side

c1

c2

c3

c4

c5

c6

(c)

Figure 4 CPU time of the standard splitting scheme and theiterative schemes with the Zassenhaus formula where we have atleast 1 4 iterative steps

International Journal of Differential Equations 9

calculating the nilpotent commutators in the Zassenhausformula So it makes sense to have 1ndash3 Zassenhaus stepsbefore starting the iterative scheme At that point only 2-3iterative steps are needed to obtainmore accurate results thanthose from the expensive standard schemes Here too weobtain the best results with the one-sided iterative schemes

Remark 15 Based on our restriction to 1198631 1198632 and Δ119909 with

(1198632minus 1198631)Δ1199092

asymp 1 we can control the delicate blowupof the computed commutators By the way if we neglectthe restriction and Δ119909 with (119863

2minus 1198631)Δ1199092

rarr infin thecommutator blows up and we lose the benefits of such analgorithmThis is a limitation of applying such an embeddedZassenhaus formula

322 Two Phase Example The next example is a moredelicate real life problem involving a multiphase transport-reaction equation We treat mobile and immobile pores in aporousmedia Such simulations aremade forwaste scenariossee [2]

We concentrate on the computational benefits of a fastcomputation of the commutators and their use for theinitialisation of the iterative splitting scheme

The equation is

1205971199051198881+ nabla sdot F119888

1= 119892 (minus119888

1+ 1198881119894119898

) minus 12058211198881 in Ω times [0 119905]

1205971199051198882+ nabla sdot F119888

2= 119892 (minus119888

2+ 1198882119894119898

) + 12058211198881minus 12058221198882 in Ω times [0 119905]

F = v minus 119863nabla

1205971199051198881119894119898

= 119892 (1198881minus 1198881119894119898

) minus 12058211198881119894119898

in Ω times [0 119905]

1205971199051198882119894119898

= 119892 (1198882minus 1198882119894119898

) + 12058211198881119894119898

minus 12058221198882119894119898

in Ω times [0 119905]

1198881(x 119905) = 119888

10(x) 119888

2(x 119905) = 119888

20(x) on Ω

1198881(x 119905) = 119888

11(x 119905) 119888

2(x 119905) = 119888

21(x 119905)

on 120597Ω times [0 119905]

1198881119894119898

(x 119905) = 0 1198882119894119898

(x 119905) = 0 on Ω

1198881119894119898 (x 119905) = 0 119888

2119894119898 (x 119905) = 0 on 120597Ω times [0 119905]

(41)

Here the parameters are V = 01 119863 = 001 1205821= 1205822= 01

and 119892 = 001We next treat the semidiscretised equation given by the

matrices

120597119905C = (

119860 minus Λ1minus 119866 0 119866 0

Λ1

119860 minus Λ2minus 119866 0 119866

119866 0 minusΛ1minus 119866 0

0 119866 Λ1

minusΛ2minus 119866

)C

(42)

where C = (c1 c2 c1119894119898 c2119894119898)119879 while c1 = (119888

11 119888

1119868) is the

solution of the first species in themobile phase in each spatialdiscretisation point (119894 = 1 119868) and the same for the othersolution vectors

We have the following two operators for the splittingmethod

119860 =119863

Δ1199092sdot (

minus2 1

1 minus2 1

d d d1 minus2 1

1 minus2

)

+VΔ119909

sdot (

1

minus1 1

d dminus1 1

minus1 1

) isin R119868times119868

(43)

where 119868 is the number of spatial pointsConsider

Λ1= (

1205821

0

0 1205821

0

d d d0 1205821

0

0 1205821

)

Λ2= (

1205822

0

0 1205822

0

d d d0 1205822

0

0 1205822

) isin R119868times119868

119866 = (

119892 0

0 119892 0

d d d0 119892 0

0 119892

) isin R119868times119868

(44)

We decouple this into the following matrices

119860 = (

119860 0 0 0

0 119860 0 0

0 0 0 0

0 0 0 0

) isin R4119868times4119868

1198611= (

minusΛ1

0 0 0

Λ1

minusΛ2

0 0

0 0 minusΛ1

0

0 0 Λ1

minusΛ2

)

1198612= (

minus119866 0 119866 0

0 minus119866 0 119866

119866 0 minus119866 0

0 119866 0 minus119866

) isin R4119868times4119868

(45)

For the operators 119860 and 119861 = 1198611+ 1198612and 119861 we apply the

iterative splitting method given in Section 21Based on the decomposition119860 is block diagonal and 119861 is

tridiagonalThe commutator is

[119860 119861] = (

0 0 119860119866 0

0 0 0 119860119866

minus119860119866 0 0 0

0 minus119860119866 0 0

) isin R4119868times4119868

(46)

10 International Journal of Differential Equations

Strang

AB

102

101

100

10minus1

10minus2

10minus3

100

Δt

c1

c2

c3

c4

c5

c6

Erro

r L1

AB Strang one-sided with B

(a)

CPU

tim

e

AB Strang one-side

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

(b)

Figure 5 Numerical errors and CPU time of the standard splittingscheme and the iterative schemes with Zassenhaus formula wherewe have at least 1 4 iterative steps

where we have a sparser operator than the main operator 119861and therefore we save computation time in computing suchoperators Further we assume that we have a bounded spatialstep size which is given by Δ119909 = 01 ≫ 0 so that we scale thesingular entry119863Δ119909

2asymp 1 and VΔ119909 asymp 1

Figure 5 presents the numerical error that is the differ-ence between the exact and the numerical solution and theCPU time required The optimal results are from the one-sided iterative schemes on the operator 119861

Remark 16 We obtain the same results as in the one-phase example because of the basically constant CPUtime for decreasing time steps of the iterative scheme weobtain optimal results when applying 1ndash3 Zassenhaus stepsTherefore with the embedded Zassenhaus formula we can

reach improved results more quickly than with the standardschemes With 2-3 more iterative steps we obtain moreaccurate results With the one-sided iterative schemes wereach the best convergence results when using the moredelicate operator 119861

4 Conclusions and Discussion

In this paper we have presented a novel splitting schemecombining the ideas of iterative and sequential schemes basedon the Zassenhaus formula Here the idea is to decouplethe expensive iterative computation of the schemes basedonly on the matrix exponential obtaining simpler embeddedZassenhaus schemes based on nilpotent commutators Suchcombinations have the benefit of reducing the computationtime because the commutators can be computed very cheaplyOn the other hand simple linear iterative steps can be donevery cheaply if they are initialised with a sufficiently accuratestarting solution and this accelerates the solvers The erroranalysis showed that these are stablemethods for higher orderschemes In the applications we were able to demonstratethe speedup from the Zassenhaus enhanced method andwere able to apply it to multiphysics applications In thefuture we will concentrate on linear and nonlinear matrix-dependent schemes that switch between noniterative anditerative schemes based on Zassenhaus products

References

[1] R E Ewing ldquoUp-scaling of biological processes andmultiphaseow in porous mediardquo in IIMA Volumes in Mathematics andIts Applications vol 295 pp 195ndash215 Springer New York NYUSA 2002

[2] J Geiser Discretization and Simulation of Systems forConvection-Diffusion-Dispersion Reactions with Applicationsin Groundwater Contamination Groundwater ModellingManagement and Contamination Nova Science MonographNew York NY USA 2008

[3] N Antonic C J van Duijn W Jager and A MikelicMultiscaleProblems in Science and Technology Challenges toMathematicalAnalysis and Perspectives Springer Berlin Germany 2002

[4] E Weinan Principle of Multiscale Modelling Cambridge Uni-versity Press Cambridge UK 2011

[5] S Blanes and F Casas ldquoOn the necessity of negative coefficientsfor operator splitting schemes of order higher than twordquoAppliedNumerical Mathematics vol 54 no 1 pp 23ndash37 2005

[6] EHansen andAOstermann ldquoHigh order splittingmethods foranalytic semigroups existrdquo BIT NumericalMathematics vol 49no 3 pp 527ndash542 2009

[7] G Strang ldquoOn the construction and comparison of differenceschemesrdquo SIAM Journal on Numerical Analysis vol 5 pp 506ndash517 1968

[8] G I Marchuk ldquoSome applicatons of splitting-up methods tothe solution of problems in mathematical physicsrdquo AplikaceMatematiky vol 1 pp 103ndash132 1968

[9] J Geiser ldquoComputing exponential for iterative splitting meth-ods algorithms and applicationsrdquo Journal of AppliedMathemat-ics vol 2011 Article ID 193781 27 pages 2011

[10] R I McLachlan and G RW Quispel ldquoSplittingmethodsrdquoActaNumerica vol 11 pp 341ndash434 2002

International Journal of Differential Equations 11

[11] J Geiser G Tanoglu and N Gucuyenen ldquoHigher order oper-ator splitting methods via Zassenhaus product formula theoryand applicationsrdquo Computers amp Mathematics with Applicationsvol 62 no 4 pp 1994ndash2015 2011

[12] G H Weiss and A A Maradudin ldquoThe Baker-Hausdorff for-mula and a problem in crystal physicsrdquo Journal of MathematicalPhysics vol 3 pp 771ndash777 1962

[13] R MWilcox ldquoExponential operators and parameter differenti-ation in quantum physicsrdquo Journal of Mathematical Physics vol8 pp 962ndash982 1967

[14] J Geiser ldquoAn iterative splitting approach for linear integro-differential equationsrdquo Applied Mathematics Letters vol 26 no11 pp 1048ndash1052 2013

[15] F Casas and A Murua ldquoAn efficient algorithm for computingthe Baker-Campbell-Hausdorff series and some of its applica-tionsrdquo Journal of Mathematical Physics vol 50 no 3 Article ID033513 2009

[16] J Geiser Iterative Splitting Methods for Differential EquationsChapmanampHallCRCNumerical Analysis and Scientific Com-puting CRC Press Boca Raton Fla USA 2011

[17] S Vandewalle Parallel Multigrid Waveform Relaxation forParabolic Problems Teubner Stuttgart Germany 1993

[18] J Geiser and G Tanoglu ldquoOperator-splitting methods viathe Zassenhaus product formulardquo Applied Mathematics andComputation vol 217 no 9 pp 4557ndash4575 2011

[19] F Bayen ldquoOn the convergence of the Zassenhaus formulardquoLetters in Mathematical Physics vol 3 no 3 pp 161ndash167 1979

[20] F Casas A Murua and M Nadinic ldquoEfficient computation ofthe Zassenhaus formulardquo Computer Physics Communicationsvol 183 no 11 pp 2386ndash2391 2012

[21] E Hairer C Lubich and GWannerGeometric Numerical Inte-gration vol 31 of Springer Series in Computational MathematicsSpringer Berlin Germany 2002

[22] H Yoshida ldquoConstruction of higher order symplectic integra-torsrdquo Physics Letters A vol 150 no 5ndash7 pp 262ndash268 1990

[23] E Faou A Ostermann and K Schratz ldquoAnalysis of exponentialsplitting methods for inhomogeneous parabolic equationsrdquohttparxivorgabs12125827

[24] J Geiser ldquoDiscretization methods with embedded analyticalsolutions for convection-diffusion dispersion-reaction equa-tions and applicationsrdquo Journal of EngineeringMathematics vol57 no 1 pp 79ndash98 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 International Journal of Differential Equations

and so there exists a new product based on the Zassenhauspolynomials as follows

Πinfin

119896=3exp (119863

119896119905119896) = Π

infin

119896=2exp (119862

119896119905119896)Πinfin

119896=2exp (119862

119896119905119896) (18)

and we obtain a higher order scheme see also [22]

Remark 6 The application of the Zassenhaus formula topartial differential equations for example second-orderparabolic evolution equations needs additional stringentregularity conditions on the operators Due to the nestedcommutators which are needed to apply the Zassenhausformula one needs at least higher regularity conditions

We need some functional analytical results on thedomains of the operators The domains of the operators haveto satisfy the following compatibility conditions for the firstZassenhaus exponents

(i) Condition for the first term 1198622of the Zassenhaus

formula

D (1198712) sub D (119860

2) capD (119860119861) capD (119861119860) capD (119861

2) (19)

where119871 = 119860+119861 andD is the domain of the operators(ii) Condition for the second term 119862

3of the Zassenhaus

formula

D (1198713) sub D (119860

3) capD (119860

2119861)

capD (119860119861119860) capD (1198601198612) capD (119861

3)

(20)

where119871 = 119860+119861 andD is the domain of the operators

Example 7 If we assume to apply the Zassenhaus formula to asecond-order diffusion equation and we assume to split eachspatial direction see the functional analytical framework[23] where the domains are given as

D (119871) = 1198672(Ω) cap 119867

1

0(Ω)

D (119860) = 119888 isin 1198712(Ω) 120597

119909119909119888 120597119909119888 isin 1198712Ω

119906 (0 119910) = 119906 (1 119910) = 0 forall119910 isin (0 1)

D (119861) = 119888 isin 1198712(Ω) 120597

119910119910119888 120597119910119888 isin 1198712Ω

119906 (119909 0) = 119906 (119909 1) = 0 forall119909 isin (0 1)

(21)

whereΩ = (0 1)2

Then we need the following regularity result

(i) condition for the first term 1198622of the Zassenhaus

formula

D (1198712) sub 119867

4(Ω) (22)

(ii) condition for the second term 1198623of the Zassenhaus

formula

D (1198713) sub 119867

6(Ω) (23)

In the example we need a very high regular domain toapply a first or second Zassenhaus exponent This is one ofthe limitations and drawbacks of applying the Zassenhausformula In our applications we restrict on applying the firstor at least the second Zassenhaus exponent We apply thethird and fourth Zassenhaus exponents to the benchmarkproblems given as matrix equations without such restrictionsto the domains see [20]

23 Embedding the Zassenhaus Expansion into an IterativeSplitting Scheme We now discuss the embedding of theZassenhaus formula into the iterative operator splittingschemes

Definition 8 We apply the Lie-Trotter splitting with theembedded Zassenhaus formula which is given as

119864ZassenComp1 (119905) = exp (119860119905) exp (119861119905)

119864ZassenComp119895 (119905) = exp (119862119895119905119895) for 119895 isin 2 119894

(24)

where the sequential Zassenhaus operators are defined as

119864Zassen119894 (119905) = Π119894

119895=1119864ZassenComp119895 (119905) (25)

and we obtain an accuracy of O(119905119894) and 119894 is the number of

Zassenhaus components

In the following theorem we present the improvement ofthe iterative splitting scheme with the Zassenhaus expansion

Theorem 9 One solves the initial value problem (1) Weassume bounded and constant operators 119860 119861

The initialisation process is done with the Zassenhausformula

119888119894(119905) = 119864

119885119886119904119904119890119899119894(119905) 1198880 (26)

where 119864119885119886119904119904119890119899119894

(119905) is given in (25)Furthermore the improved solutions are embedded in the

iterative splitting scheme (4) and one has after 119895 iterative stepsthe following result

119888119894+119895

(119905) = 119864119894119905119890119903119895

(119905) 119864119885119886119904119904119890119899119894

(119905) 1198880 (27)

This has improved the error of the iterative scheme to O(119905119894+119895

)

Proof The solution of the iterative splitting scheme (4) is

119888119894+119895

(119905) = 119864iter119894 (119905) 119888init119895 (28)

where 119878119894(119905) = 119864iter119894(119905) given inTheorem 1

The initialisation is given by the Zassenhaus formula

119888init119895 (119905) = 119864Zassen119895 (119905) 1198880 (29)

Combining both splitting schemes we have a local error of

119890119894+119895 (119905) =

10038171003817100381710038171003817119888 (119905) minus 119888

119894+119895 (119905)10038171003817100381710038171003817

=10038171003817100381710038171003817119888 (119905) minus 119864iter119894 (119905) 119864Zassen119895 (119905) 1198880

10038171003817100381710038171003817

le O (119905119894+119895

) 1198880

(30)

International Journal of Differential Equations 5

Remark 10 We obtain an acceleration of the iterative schemeby applying cheap higher order Zassenhaus formula Such abenefit allows reducing to 2-3 iterative steps while the initialsolution is of a higher order Taking into account the sparsityor nilpotency of the commutators we nowhave a fast iterativesplitting scheme

Remark 11 For our numerical experiment we apply theZassenhaus formulaat the beginning of each integration stepas the so-called preprocessor Therefore we obtain moreaccurate starting conditions for the integration steps of theiterative splitting scheme Based on the regularity conditionsof the nested commutator we apply only 1-2 Zassenhausexponents

In the following we explain the final algorithm ofthe embedded Zassenhaus formula to the iterative splittingscheme

Algorithm 12 We compute 119873 time steps where each timestep Δ119905

119899is given as Δ119905

119899= 119905119899+1

minus 119905119899 with 119899 = 0 119873 minus 1

Further 119888(1199050) is the initial condition of the Cauchy problem

(1) and we compute 119888(1199051) 119888(119905119873) successively We have thefollowing two steps first we apply the Zassenhaus formulaas initialization of 119888

0(119905119899+1

) for the iterative scheme secondwe apply the iterative splitting scheme to the initial valueproblem (1) and compute the final solution 119888(119905

119899+1) as follows

(1) We start with 119895 = 2 and compute the Zassenhausformula

(2) We compute the matrix norm for the Zassenhaussexponents 119862

119895

If errZass119895

le err1 then we start with the iterative

splitting scheme with 1198880(119905) = 119888Zass

119895

119894 = 1 and go to(3) else 119895 = 119895 + 1 and we go to (1)

(3) If the error of the iterative scheme erriter le err2 then

we are finished else 119894 = 119894+1 and we compute the nextiterative step till erriter le err

2

3 Numerical Examples

In this section we discuss examples of the use of embeddingZassenhaus product methods in iterative splitting schemesIn the first example we demonstrate somewhat artificiallyhow the proposed Zassenhaus splitting method avoids thesplitting error up to a certain order The next example showsthe benefit of the combined splitting schemes for applicationsto multiphysics problems

In the following we consider a benchmark example tovalidate the theoretical results and real life problems to applyour schemes

We applied the following splitting schemes given inTable 1

31 Benchmark Problem Matrix Examples In the followingthe idea of thematrix example is to present the decompositioninto two simpler matrices In context of the multiphysics

Table 1 Applied splitting schemes

Abbreviation inthe examples Splitting methods

A-B A-B splittingStrang Strang splitting1198881

Iterative splitting with one iterative step

1198882

Iterative splitting with one iterative step and oneZassenhaus step as prestep

1198883

Iterative splitting with one iterative step and twoZassenhaus steps as presteps

1198884

Iterative splitting with two iterative steps andtwo Zassenhaus steps as presteps

1198885

Iterative splitting with three iterative steps andtwo Zassenhaus steps as presteps

1198886

Iterative splitting with four iterative steps andtwo Zassenhaus steps as presteps

problems even such simple problems can be seen as testexamples We assume that each simple matrix is presenting abasic single physics problems for example reaction processbetween two species while the combination of the twomatrices results in a more complicate physics problem forexample coupled reaction process between two species see[2]

We consider the matrix equation

1199061015840(119905) = [

1 2

3 0] 119906 119906 (0) = 119906

0= (

0

1) 119905 isin [0 1] (31)

the exact solution of which is

119906 (119905) =2 (1198903119905

minus 119890minus2119905

)

5 (32)

We split the matrix as

[1 2

3 0] = [

1 1

1 0] + [

0 1

2 0] = 119860 + 119861 (33)

where [119860 119861] = 0 For this simple example we did not obtainnilpotent matrices for the commutators

We apply 10ndash100 time steps for the computations Thefinal integration time of each single time step was less thanlt1 [sec] such that we need at least for such benchmarkproblems about 1ndash10 [sec] Figure 1 presents the numericalerrors that is the difference between the exact and thenumerical solution

Figure 2 presents the CPU times needed by the standardand the iterative splitting schemes The application of theZassenhaus formula did not increase while the next Zassen-haus coefficients are at least nearly nilpotent matrices and wecan neglect the computational timeMoreover different time-steps did not influence the computations of the Zassenhausexponents which is a preprocess First we compute and saveall the Zassenhaus exponents and later we multiply with thenecessary time steps

6 International Journal of Differential Equations

102

102

10minus2

10minus4

10minus6

10minus8

10minus2

10minus1

100

AB Strang one-sided

AB

Δt

Strang

c1

c2

c3

c4

c5

c6

Erro

r L1

(a)

AB

Δt

Strang

102

102

10minus2

10minus4

10minus6

10minus8

10minus2

10minus1

100

AB Strang two-sided

c1

c2

c3

c4

c5

c6

Erro

r L1

(b)

Figure 1 Numerical errors of the standard splitting scheme and theiterative schemes with the Zassenhaus formula where we have atleast 1 4 iterative steps

Remark 13 We see that the errors decrease significantlywith increasing order of approximation for the initialisa-tion process from the Zassenhaus splitting (we apply 1ndash3Zassenhaus coefficients) Here we apply commutators whichdid not reduce their matrix rank therefore we have thesame computation time as for the pure iterative schemesOverall we have their benefits in the application of theZassenhaus product schemes to the standard Lie-Trotter orStrang-Marchuk splitting Similar results are given withmoreiterative steps 119894 = 3 6 as expected

32 Real Life Problems Multiphase Examples In the nextexamples we simulate coupled transport and reaction equa-tions with different phases so-called multiphase problems

CPU

tim

e

AB Strang one-sided

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

(a)

CPU

tim

e

AB Strang two-sided

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

(b)

Figure 2 CPU times of the standard splitting scheme and theiterative schemes with the Zassenhaus formula where we have atleast 1 4 iterative steps

see [9]The equations are coupledwith the reaction terms andare presented as follows

120597119905119877119894119906119894+ nabla sdot v119906

119894= minus 120582

119894119877119894119906119894+ 120582119894minus1

119877119894minus1

119906119894minus1

+ 120573 (minus119906119894+ 119892119894) inΩ times (0 119879)

1199061198940

(119909) = 119906119894(119909 0) on Ω

120597119905119877119894119892119894= minus 120582

119894119877119894119892119894+ 120582119894minus1

119877119894minus1

119892119894minus1

+ 120573 (minus119892119894+ 119906119894) in Ω times (0 119879)

1198921198940

(119909) = 119892119894(119909 0) on Ω

119894 = 1 119898

(34)

International Journal of Differential Equations 7

where 119898 is the number of equations and 119894 is the index ofeach component The unknown mobile concentrations 119906

119894=

119906119894(119909 119905) are taken in Ω times (0 119879) sub R119899 times R+ where 119899 is the

spatial dimension The unknown immobile concentrations119892119894= 119892119894(119909 119905) are taken in Ω times (0 119879) sub R119899 times R+ where 119899 is

the spatial dimensionThe retardation factors119877119894are constant

and 119877119894

ge 0 The kinetic part is given by the factors 120582119894

They are constant and 120582119894ge 0 For the initialisation of the

kinetic part we set 1205820

= 0 The kinetic part is linear andirreversible so the successors have only one predecessor Theinitial conditions are given for each component 119894 as constantsor linear impulses For the boundary conditions we havetrivial inflow and outflow conditions with 119906

119894= 0 at the inflow

boundary The transport part is given by the velocity v isin R119899

and is piecewise constant see [2] The exchange between themobile and immobile part is given by 120573

In the following we discuss the one-phase and two-phaseexamples

321 One-Phase Example The next example is a simplifiedreal life problem with a multiphase transport-reaction equa-tion We treat the mobile and immobile pores in a porousmedia Such simulations are made about waste scenarios see[2 24]

We concentrate on the computational benefits of a fastcomputation of the mixed iterative scheme with the Zassen-haus formula

The one phase equation is

1205971199051198881+ nabla sdot F

11198881= minus12058211198881 in Ω times [0 119905]

1205971199051198882+ nabla sdot F

21198882= 12058211198881minus 12058221198882 in Ω times [0 119905]

F119894= v119894minus 119863119894nabla 119894 = 1 2

1198881(x 119905) = 119888

10(x) 119888

2(x 119905) = 119888

20(x) on Ω

1198881 (x 119905) = 119888

11 (x 119905) 1198882 (x 119905) = 119888

21 (x 119905)

on 120597Ω times [0 119905]

(35)

Here the parameters take the values V1= 01 V

2= 005119863

1=

0011198632= 0005 and 120582

1= 1205822= 01

We now turn to the finite difference schemes andsemidiscretise the equation with its convection and diffusionoperators as follows

120597119905c = (119860

1+ 1198602) c (36)

We obtain the two matrices and decouple the diffusion andconvection parts as follows

1198601= (

119860diff 0

0 119860diff) isin R

2119868times2119868

1198602= (

119860Conv 0

0 119860Conv) + (

minusΛ1

0

Λ1

minusΛ2

) isin R2119868times2119868

(37)

We apply the splitting method to the operators 1198601and 119860

2

given in Section 21

The submatrices are

119860diff119894 =119863119894

Δ11990921198771=

119863119894

Δ1199092sdot (

minus2 1

1 minus2 1

d d d1 minus2 1

1 minus2

) isin R119868times119868

119860conv119894 = minusV119894

Δ1199091198772= minus

V119894

Δ119909sdot (

1

minus1 1

d dminus1 1

minus1 1

) isin R119868times119868

(38)

where 119868 is the number of spatial points and Δ119909 is the spatialstep size Consider

Λ1= (

1205821

0

0 1205821

0

d d d0 1205821

0

0 1205821

) isin R119868times119868

Λ2= (

1205822

0

0 1205822

0

d d d0 1205822

0

0 1205822

) isin R119868times119868

(39)

We have the commutator

[1198601 1198602] = (

0 0

1198632minus 1198631

Δ11990921198771Λ1

0) isin R

2119868times2119868 (40)

where we assume that [1198771 1198772] asymp 0 This is a nilpotent

commutator and the computation time needed for thecommutators of such operators is less than that for the fulloperators Furthermore we assume that we have a boundedspatial step size which is given by Δ119909 = 01 ≫ 0 so thatwe scale the singular entry (119863

2minus 1198631)Δ1199092asymp 1 and evade a

blowup in such matricesWe have obtained the optimal results of a basically

constant CPU time for decreasing time steps which increasein the standard scheme

We apply 10ndash100 timesteps for the computations with 10ndash100 spatial points The final integration time of each singletime step was about 1ndash10 [sec] such that we need at least forsuch benchmark problems about 100ndash1000 [sec]

Figure 3 presents the numerical error that is the differ-ence between the exact and the numerical solution

Figure 4 presents the CPU time of the standard andthe iterative splitting schemes Based on the preprocess ofcomputing the Zassenhaus exponents at the beginning andstore the matrices we could save additional computationaltime with larger time steps

Remark 14 For the iterative schemes with embedded Zassen-haus products we can reach improved results more quicklyThe benefits come from the constant CPU time needed for

8 International Journal of Differential Equations

AB

Δt

Strang

101

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

Erro

r L1

AB Strang one-sided with A

(a)

AB

Δt

Strang

102

100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

Erro

r L1

AB Strang one-sided with B

(b)

AB

Δt

Strang

102

100

10minus2

10minus4

10minus6

10minus8

10minus2

10minus1

100

AB Strang two-side

c1

c2

c3

c4

c5

c6

Erro

r L1

(c)

Figure 3 Numerical errors of the standard splitting scheme and theiterative schemes with the Zassenhaus formula where we have atleast 1 4 iterative steps

CPU

tim

e

AB strang one-side with A

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

(a)

CPU

tim

e

Δt

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

AB

Strang

c1

c2

c3

c4

c5

c6

AB Strang one-sided with B

(b)

CPU

tim

e

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

AB Strang two-side

c1

c2

c3

c4

c5

c6

(c)

Figure 4 CPU time of the standard splitting scheme and theiterative schemes with the Zassenhaus formula where we have atleast 1 4 iterative steps

International Journal of Differential Equations 9

calculating the nilpotent commutators in the Zassenhausformula So it makes sense to have 1ndash3 Zassenhaus stepsbefore starting the iterative scheme At that point only 2-3iterative steps are needed to obtainmore accurate results thanthose from the expensive standard schemes Here too weobtain the best results with the one-sided iterative schemes

Remark 15 Based on our restriction to 1198631 1198632 and Δ119909 with

(1198632minus 1198631)Δ1199092

asymp 1 we can control the delicate blowupof the computed commutators By the way if we neglectthe restriction and Δ119909 with (119863

2minus 1198631)Δ1199092

rarr infin thecommutator blows up and we lose the benefits of such analgorithmThis is a limitation of applying such an embeddedZassenhaus formula

322 Two Phase Example The next example is a moredelicate real life problem involving a multiphase transport-reaction equation We treat mobile and immobile pores in aporousmedia Such simulations aremade forwaste scenariossee [2]

We concentrate on the computational benefits of a fastcomputation of the commutators and their use for theinitialisation of the iterative splitting scheme

The equation is

1205971199051198881+ nabla sdot F119888

1= 119892 (minus119888

1+ 1198881119894119898

) minus 12058211198881 in Ω times [0 119905]

1205971199051198882+ nabla sdot F119888

2= 119892 (minus119888

2+ 1198882119894119898

) + 12058211198881minus 12058221198882 in Ω times [0 119905]

F = v minus 119863nabla

1205971199051198881119894119898

= 119892 (1198881minus 1198881119894119898

) minus 12058211198881119894119898

in Ω times [0 119905]

1205971199051198882119894119898

= 119892 (1198882minus 1198882119894119898

) + 12058211198881119894119898

minus 12058221198882119894119898

in Ω times [0 119905]

1198881(x 119905) = 119888

10(x) 119888

2(x 119905) = 119888

20(x) on Ω

1198881(x 119905) = 119888

11(x 119905) 119888

2(x 119905) = 119888

21(x 119905)

on 120597Ω times [0 119905]

1198881119894119898

(x 119905) = 0 1198882119894119898

(x 119905) = 0 on Ω

1198881119894119898 (x 119905) = 0 119888

2119894119898 (x 119905) = 0 on 120597Ω times [0 119905]

(41)

Here the parameters are V = 01 119863 = 001 1205821= 1205822= 01

and 119892 = 001We next treat the semidiscretised equation given by the

matrices

120597119905C = (

119860 minus Λ1minus 119866 0 119866 0

Λ1

119860 minus Λ2minus 119866 0 119866

119866 0 minusΛ1minus 119866 0

0 119866 Λ1

minusΛ2minus 119866

)C

(42)

where C = (c1 c2 c1119894119898 c2119894119898)119879 while c1 = (119888

11 119888

1119868) is the

solution of the first species in themobile phase in each spatialdiscretisation point (119894 = 1 119868) and the same for the othersolution vectors

We have the following two operators for the splittingmethod

119860 =119863

Δ1199092sdot (

minus2 1

1 minus2 1

d d d1 minus2 1

1 minus2

)

+VΔ119909

sdot (

1

minus1 1

d dminus1 1

minus1 1

) isin R119868times119868

(43)

where 119868 is the number of spatial pointsConsider

Λ1= (

1205821

0

0 1205821

0

d d d0 1205821

0

0 1205821

)

Λ2= (

1205822

0

0 1205822

0

d d d0 1205822

0

0 1205822

) isin R119868times119868

119866 = (

119892 0

0 119892 0

d d d0 119892 0

0 119892

) isin R119868times119868

(44)

We decouple this into the following matrices

119860 = (

119860 0 0 0

0 119860 0 0

0 0 0 0

0 0 0 0

) isin R4119868times4119868

1198611= (

minusΛ1

0 0 0

Λ1

minusΛ2

0 0

0 0 minusΛ1

0

0 0 Λ1

minusΛ2

)

1198612= (

minus119866 0 119866 0

0 minus119866 0 119866

119866 0 minus119866 0

0 119866 0 minus119866

) isin R4119868times4119868

(45)

For the operators 119860 and 119861 = 1198611+ 1198612and 119861 we apply the

iterative splitting method given in Section 21Based on the decomposition119860 is block diagonal and 119861 is

tridiagonalThe commutator is

[119860 119861] = (

0 0 119860119866 0

0 0 0 119860119866

minus119860119866 0 0 0

0 minus119860119866 0 0

) isin R4119868times4119868

(46)

10 International Journal of Differential Equations

Strang

AB

102

101

100

10minus1

10minus2

10minus3

100

Δt

c1

c2

c3

c4

c5

c6

Erro

r L1

AB Strang one-sided with B

(a)

CPU

tim

e

AB Strang one-side

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

(b)

Figure 5 Numerical errors and CPU time of the standard splittingscheme and the iterative schemes with Zassenhaus formula wherewe have at least 1 4 iterative steps

where we have a sparser operator than the main operator 119861and therefore we save computation time in computing suchoperators Further we assume that we have a bounded spatialstep size which is given by Δ119909 = 01 ≫ 0 so that we scale thesingular entry119863Δ119909

2asymp 1 and VΔ119909 asymp 1

Figure 5 presents the numerical error that is the differ-ence between the exact and the numerical solution and theCPU time required The optimal results are from the one-sided iterative schemes on the operator 119861

Remark 16 We obtain the same results as in the one-phase example because of the basically constant CPUtime for decreasing time steps of the iterative scheme weobtain optimal results when applying 1ndash3 Zassenhaus stepsTherefore with the embedded Zassenhaus formula we can

reach improved results more quickly than with the standardschemes With 2-3 more iterative steps we obtain moreaccurate results With the one-sided iterative schemes wereach the best convergence results when using the moredelicate operator 119861

4 Conclusions and Discussion

In this paper we have presented a novel splitting schemecombining the ideas of iterative and sequential schemes basedon the Zassenhaus formula Here the idea is to decouplethe expensive iterative computation of the schemes basedonly on the matrix exponential obtaining simpler embeddedZassenhaus schemes based on nilpotent commutators Suchcombinations have the benefit of reducing the computationtime because the commutators can be computed very cheaplyOn the other hand simple linear iterative steps can be donevery cheaply if they are initialised with a sufficiently accuratestarting solution and this accelerates the solvers The erroranalysis showed that these are stablemethods for higher orderschemes In the applications we were able to demonstratethe speedup from the Zassenhaus enhanced method andwere able to apply it to multiphysics applications In thefuture we will concentrate on linear and nonlinear matrix-dependent schemes that switch between noniterative anditerative schemes based on Zassenhaus products

References

[1] R E Ewing ldquoUp-scaling of biological processes andmultiphaseow in porous mediardquo in IIMA Volumes in Mathematics andIts Applications vol 295 pp 195ndash215 Springer New York NYUSA 2002

[2] J Geiser Discretization and Simulation of Systems forConvection-Diffusion-Dispersion Reactions with Applicationsin Groundwater Contamination Groundwater ModellingManagement and Contamination Nova Science MonographNew York NY USA 2008

[3] N Antonic C J van Duijn W Jager and A MikelicMultiscaleProblems in Science and Technology Challenges toMathematicalAnalysis and Perspectives Springer Berlin Germany 2002

[4] E Weinan Principle of Multiscale Modelling Cambridge Uni-versity Press Cambridge UK 2011

[5] S Blanes and F Casas ldquoOn the necessity of negative coefficientsfor operator splitting schemes of order higher than twordquoAppliedNumerical Mathematics vol 54 no 1 pp 23ndash37 2005

[6] EHansen andAOstermann ldquoHigh order splittingmethods foranalytic semigroups existrdquo BIT NumericalMathematics vol 49no 3 pp 527ndash542 2009

[7] G Strang ldquoOn the construction and comparison of differenceschemesrdquo SIAM Journal on Numerical Analysis vol 5 pp 506ndash517 1968

[8] G I Marchuk ldquoSome applicatons of splitting-up methods tothe solution of problems in mathematical physicsrdquo AplikaceMatematiky vol 1 pp 103ndash132 1968

[9] J Geiser ldquoComputing exponential for iterative splitting meth-ods algorithms and applicationsrdquo Journal of AppliedMathemat-ics vol 2011 Article ID 193781 27 pages 2011

[10] R I McLachlan and G RW Quispel ldquoSplittingmethodsrdquoActaNumerica vol 11 pp 341ndash434 2002

International Journal of Differential Equations 11

[11] J Geiser G Tanoglu and N Gucuyenen ldquoHigher order oper-ator splitting methods via Zassenhaus product formula theoryand applicationsrdquo Computers amp Mathematics with Applicationsvol 62 no 4 pp 1994ndash2015 2011

[12] G H Weiss and A A Maradudin ldquoThe Baker-Hausdorff for-mula and a problem in crystal physicsrdquo Journal of MathematicalPhysics vol 3 pp 771ndash777 1962

[13] R MWilcox ldquoExponential operators and parameter differenti-ation in quantum physicsrdquo Journal of Mathematical Physics vol8 pp 962ndash982 1967

[14] J Geiser ldquoAn iterative splitting approach for linear integro-differential equationsrdquo Applied Mathematics Letters vol 26 no11 pp 1048ndash1052 2013

[15] F Casas and A Murua ldquoAn efficient algorithm for computingthe Baker-Campbell-Hausdorff series and some of its applica-tionsrdquo Journal of Mathematical Physics vol 50 no 3 Article ID033513 2009

[16] J Geiser Iterative Splitting Methods for Differential EquationsChapmanampHallCRCNumerical Analysis and Scientific Com-puting CRC Press Boca Raton Fla USA 2011

[17] S Vandewalle Parallel Multigrid Waveform Relaxation forParabolic Problems Teubner Stuttgart Germany 1993

[18] J Geiser and G Tanoglu ldquoOperator-splitting methods viathe Zassenhaus product formulardquo Applied Mathematics andComputation vol 217 no 9 pp 4557ndash4575 2011

[19] F Bayen ldquoOn the convergence of the Zassenhaus formulardquoLetters in Mathematical Physics vol 3 no 3 pp 161ndash167 1979

[20] F Casas A Murua and M Nadinic ldquoEfficient computation ofthe Zassenhaus formulardquo Computer Physics Communicationsvol 183 no 11 pp 2386ndash2391 2012

[21] E Hairer C Lubich and GWannerGeometric Numerical Inte-gration vol 31 of Springer Series in Computational MathematicsSpringer Berlin Germany 2002

[22] H Yoshida ldquoConstruction of higher order symplectic integra-torsrdquo Physics Letters A vol 150 no 5ndash7 pp 262ndash268 1990

[23] E Faou A Ostermann and K Schratz ldquoAnalysis of exponentialsplitting methods for inhomogeneous parabolic equationsrdquohttparxivorgabs12125827

[24] J Geiser ldquoDiscretization methods with embedded analyticalsolutions for convection-diffusion dispersion-reaction equa-tions and applicationsrdquo Journal of EngineeringMathematics vol57 no 1 pp 79ndash98 2007

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Differential Equations 5

Remark 10 We obtain an acceleration of the iterative schemeby applying cheap higher order Zassenhaus formula Such abenefit allows reducing to 2-3 iterative steps while the initialsolution is of a higher order Taking into account the sparsityor nilpotency of the commutators we nowhave a fast iterativesplitting scheme

Remark 11 For our numerical experiment we apply theZassenhaus formulaat the beginning of each integration stepas the so-called preprocessor Therefore we obtain moreaccurate starting conditions for the integration steps of theiterative splitting scheme Based on the regularity conditionsof the nested commutator we apply only 1-2 Zassenhausexponents

In the following we explain the final algorithm ofthe embedded Zassenhaus formula to the iterative splittingscheme

Algorithm 12 We compute 119873 time steps where each timestep Δ119905

119899is given as Δ119905

119899= 119905119899+1

minus 119905119899 with 119899 = 0 119873 minus 1

Further 119888(1199050) is the initial condition of the Cauchy problem

(1) and we compute 119888(1199051) 119888(119905119873) successively We have thefollowing two steps first we apply the Zassenhaus formulaas initialization of 119888

0(119905119899+1

) for the iterative scheme secondwe apply the iterative splitting scheme to the initial valueproblem (1) and compute the final solution 119888(119905

119899+1) as follows

(1) We start with 119895 = 2 and compute the Zassenhausformula

(2) We compute the matrix norm for the Zassenhaussexponents 119862

119895

If errZass119895

le err1 then we start with the iterative

splitting scheme with 1198880(119905) = 119888Zass

119895

119894 = 1 and go to(3) else 119895 = 119895 + 1 and we go to (1)

(3) If the error of the iterative scheme erriter le err2 then

we are finished else 119894 = 119894+1 and we compute the nextiterative step till erriter le err

2

3 Numerical Examples

In this section we discuss examples of the use of embeddingZassenhaus product methods in iterative splitting schemesIn the first example we demonstrate somewhat artificiallyhow the proposed Zassenhaus splitting method avoids thesplitting error up to a certain order The next example showsthe benefit of the combined splitting schemes for applicationsto multiphysics problems

In the following we consider a benchmark example tovalidate the theoretical results and real life problems to applyour schemes

We applied the following splitting schemes given inTable 1

31 Benchmark Problem Matrix Examples In the followingthe idea of thematrix example is to present the decompositioninto two simpler matrices In context of the multiphysics

Table 1 Applied splitting schemes

Abbreviation inthe examples Splitting methods

A-B A-B splittingStrang Strang splitting1198881

Iterative splitting with one iterative step

1198882

Iterative splitting with one iterative step and oneZassenhaus step as prestep

1198883

Iterative splitting with one iterative step and twoZassenhaus steps as presteps

1198884

Iterative splitting with two iterative steps andtwo Zassenhaus steps as presteps

1198885

Iterative splitting with three iterative steps andtwo Zassenhaus steps as presteps

1198886

Iterative splitting with four iterative steps andtwo Zassenhaus steps as presteps

problems even such simple problems can be seen as testexamples We assume that each simple matrix is presenting abasic single physics problems for example reaction processbetween two species while the combination of the twomatrices results in a more complicate physics problem forexample coupled reaction process between two species see[2]

We consider the matrix equation

1199061015840(119905) = [

1 2

3 0] 119906 119906 (0) = 119906

0= (

0

1) 119905 isin [0 1] (31)

the exact solution of which is

119906 (119905) =2 (1198903119905

minus 119890minus2119905

)

5 (32)

We split the matrix as

[1 2

3 0] = [

1 1

1 0] + [

0 1

2 0] = 119860 + 119861 (33)

where [119860 119861] = 0 For this simple example we did not obtainnilpotent matrices for the commutators

We apply 10ndash100 time steps for the computations Thefinal integration time of each single time step was less thanlt1 [sec] such that we need at least for such benchmarkproblems about 1ndash10 [sec] Figure 1 presents the numericalerrors that is the difference between the exact and thenumerical solution

Figure 2 presents the CPU times needed by the standardand the iterative splitting schemes The application of theZassenhaus formula did not increase while the next Zassen-haus coefficients are at least nearly nilpotent matrices and wecan neglect the computational timeMoreover different time-steps did not influence the computations of the Zassenhausexponents which is a preprocess First we compute and saveall the Zassenhaus exponents and later we multiply with thenecessary time steps

6 International Journal of Differential Equations

102

102

10minus2

10minus4

10minus6

10minus8

10minus2

10minus1

100

AB Strang one-sided

AB

Δt

Strang

c1

c2

c3

c4

c5

c6

Erro

r L1

(a)

AB

Δt

Strang

102

102

10minus2

10minus4

10minus6

10minus8

10minus2

10minus1

100

AB Strang two-sided

c1

c2

c3

c4

c5

c6

Erro

r L1

(b)

Figure 1 Numerical errors of the standard splitting scheme and theiterative schemes with the Zassenhaus formula where we have atleast 1 4 iterative steps

Remark 13 We see that the errors decrease significantlywith increasing order of approximation for the initialisa-tion process from the Zassenhaus splitting (we apply 1ndash3Zassenhaus coefficients) Here we apply commutators whichdid not reduce their matrix rank therefore we have thesame computation time as for the pure iterative schemesOverall we have their benefits in the application of theZassenhaus product schemes to the standard Lie-Trotter orStrang-Marchuk splitting Similar results are given withmoreiterative steps 119894 = 3 6 as expected

32 Real Life Problems Multiphase Examples In the nextexamples we simulate coupled transport and reaction equa-tions with different phases so-called multiphase problems

CPU

tim

e

AB Strang one-sided

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

(a)

CPU

tim

e

AB Strang two-sided

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

(b)

Figure 2 CPU times of the standard splitting scheme and theiterative schemes with the Zassenhaus formula where we have atleast 1 4 iterative steps

see [9]The equations are coupledwith the reaction terms andare presented as follows

120597119905119877119894119906119894+ nabla sdot v119906

119894= minus 120582

119894119877119894119906119894+ 120582119894minus1

119877119894minus1

119906119894minus1

+ 120573 (minus119906119894+ 119892119894) inΩ times (0 119879)

1199061198940

(119909) = 119906119894(119909 0) on Ω

120597119905119877119894119892119894= minus 120582

119894119877119894119892119894+ 120582119894minus1

119877119894minus1

119892119894minus1

+ 120573 (minus119892119894+ 119906119894) in Ω times (0 119879)

1198921198940

(119909) = 119892119894(119909 0) on Ω

119894 = 1 119898

(34)

International Journal of Differential Equations 7

where 119898 is the number of equations and 119894 is the index ofeach component The unknown mobile concentrations 119906

119894=

119906119894(119909 119905) are taken in Ω times (0 119879) sub R119899 times R+ where 119899 is the

spatial dimension The unknown immobile concentrations119892119894= 119892119894(119909 119905) are taken in Ω times (0 119879) sub R119899 times R+ where 119899 is

the spatial dimensionThe retardation factors119877119894are constant

and 119877119894

ge 0 The kinetic part is given by the factors 120582119894

They are constant and 120582119894ge 0 For the initialisation of the

kinetic part we set 1205820

= 0 The kinetic part is linear andirreversible so the successors have only one predecessor Theinitial conditions are given for each component 119894 as constantsor linear impulses For the boundary conditions we havetrivial inflow and outflow conditions with 119906

119894= 0 at the inflow

boundary The transport part is given by the velocity v isin R119899

and is piecewise constant see [2] The exchange between themobile and immobile part is given by 120573

In the following we discuss the one-phase and two-phaseexamples

321 One-Phase Example The next example is a simplifiedreal life problem with a multiphase transport-reaction equa-tion We treat the mobile and immobile pores in a porousmedia Such simulations are made about waste scenarios see[2 24]

We concentrate on the computational benefits of a fastcomputation of the mixed iterative scheme with the Zassen-haus formula

The one phase equation is

1205971199051198881+ nabla sdot F

11198881= minus12058211198881 in Ω times [0 119905]

1205971199051198882+ nabla sdot F

21198882= 12058211198881minus 12058221198882 in Ω times [0 119905]

F119894= v119894minus 119863119894nabla 119894 = 1 2

1198881(x 119905) = 119888

10(x) 119888

2(x 119905) = 119888

20(x) on Ω

1198881 (x 119905) = 119888

11 (x 119905) 1198882 (x 119905) = 119888

21 (x 119905)

on 120597Ω times [0 119905]

(35)

Here the parameters take the values V1= 01 V

2= 005119863

1=

0011198632= 0005 and 120582

1= 1205822= 01

We now turn to the finite difference schemes andsemidiscretise the equation with its convection and diffusionoperators as follows

120597119905c = (119860

1+ 1198602) c (36)

We obtain the two matrices and decouple the diffusion andconvection parts as follows

1198601= (

119860diff 0

0 119860diff) isin R

2119868times2119868

1198602= (

119860Conv 0

0 119860Conv) + (

minusΛ1

0

Λ1

minusΛ2

) isin R2119868times2119868

(37)

We apply the splitting method to the operators 1198601and 119860

2

given in Section 21

The submatrices are

119860diff119894 =119863119894

Δ11990921198771=

119863119894

Δ1199092sdot (

minus2 1

1 minus2 1

d d d1 minus2 1

1 minus2

) isin R119868times119868

119860conv119894 = minusV119894

Δ1199091198772= minus

V119894

Δ119909sdot (

1

minus1 1

d dminus1 1

minus1 1

) isin R119868times119868

(38)

where 119868 is the number of spatial points and Δ119909 is the spatialstep size Consider

Λ1= (

1205821

0

0 1205821

0

d d d0 1205821

0

0 1205821

) isin R119868times119868

Λ2= (

1205822

0

0 1205822

0

d d d0 1205822

0

0 1205822

) isin R119868times119868

(39)

We have the commutator

[1198601 1198602] = (

0 0

1198632minus 1198631

Δ11990921198771Λ1

0) isin R

2119868times2119868 (40)

where we assume that [1198771 1198772] asymp 0 This is a nilpotent

commutator and the computation time needed for thecommutators of such operators is less than that for the fulloperators Furthermore we assume that we have a boundedspatial step size which is given by Δ119909 = 01 ≫ 0 so thatwe scale the singular entry (119863

2minus 1198631)Δ1199092asymp 1 and evade a

blowup in such matricesWe have obtained the optimal results of a basically

constant CPU time for decreasing time steps which increasein the standard scheme

We apply 10ndash100 timesteps for the computations with 10ndash100 spatial points The final integration time of each singletime step was about 1ndash10 [sec] such that we need at least forsuch benchmark problems about 100ndash1000 [sec]

Figure 3 presents the numerical error that is the differ-ence between the exact and the numerical solution

Figure 4 presents the CPU time of the standard andthe iterative splitting schemes Based on the preprocess ofcomputing the Zassenhaus exponents at the beginning andstore the matrices we could save additional computationaltime with larger time steps

Remark 14 For the iterative schemes with embedded Zassen-haus products we can reach improved results more quicklyThe benefits come from the constant CPU time needed for

8 International Journal of Differential Equations

AB

Δt

Strang

101

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

Erro

r L1

AB Strang one-sided with A

(a)

AB

Δt

Strang

102

100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

Erro

r L1

AB Strang one-sided with B

(b)

AB

Δt

Strang

102

100

10minus2

10minus4

10minus6

10minus8

10minus2

10minus1

100

AB Strang two-side

c1

c2

c3

c4

c5

c6

Erro

r L1

(c)

Figure 3 Numerical errors of the standard splitting scheme and theiterative schemes with the Zassenhaus formula where we have atleast 1 4 iterative steps

CPU

tim

e

AB strang one-side with A

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

(a)

CPU

tim

e

Δt

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

AB

Strang

c1

c2

c3

c4

c5

c6

AB Strang one-sided with B

(b)

CPU

tim

e

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

AB Strang two-side

c1

c2

c3

c4

c5

c6

(c)

Figure 4 CPU time of the standard splitting scheme and theiterative schemes with the Zassenhaus formula where we have atleast 1 4 iterative steps

International Journal of Differential Equations 9

calculating the nilpotent commutators in the Zassenhausformula So it makes sense to have 1ndash3 Zassenhaus stepsbefore starting the iterative scheme At that point only 2-3iterative steps are needed to obtainmore accurate results thanthose from the expensive standard schemes Here too weobtain the best results with the one-sided iterative schemes

Remark 15 Based on our restriction to 1198631 1198632 and Δ119909 with

(1198632minus 1198631)Δ1199092

asymp 1 we can control the delicate blowupof the computed commutators By the way if we neglectthe restriction and Δ119909 with (119863

2minus 1198631)Δ1199092

rarr infin thecommutator blows up and we lose the benefits of such analgorithmThis is a limitation of applying such an embeddedZassenhaus formula

322 Two Phase Example The next example is a moredelicate real life problem involving a multiphase transport-reaction equation We treat mobile and immobile pores in aporousmedia Such simulations aremade forwaste scenariossee [2]

We concentrate on the computational benefits of a fastcomputation of the commutators and their use for theinitialisation of the iterative splitting scheme

The equation is

1205971199051198881+ nabla sdot F119888

1= 119892 (minus119888

1+ 1198881119894119898

) minus 12058211198881 in Ω times [0 119905]

1205971199051198882+ nabla sdot F119888

2= 119892 (minus119888

2+ 1198882119894119898

) + 12058211198881minus 12058221198882 in Ω times [0 119905]

F = v minus 119863nabla

1205971199051198881119894119898

= 119892 (1198881minus 1198881119894119898

) minus 12058211198881119894119898

in Ω times [0 119905]

1205971199051198882119894119898

= 119892 (1198882minus 1198882119894119898

) + 12058211198881119894119898

minus 12058221198882119894119898

in Ω times [0 119905]

1198881(x 119905) = 119888

10(x) 119888

2(x 119905) = 119888

20(x) on Ω

1198881(x 119905) = 119888

11(x 119905) 119888

2(x 119905) = 119888

21(x 119905)

on 120597Ω times [0 119905]

1198881119894119898

(x 119905) = 0 1198882119894119898

(x 119905) = 0 on Ω

1198881119894119898 (x 119905) = 0 119888

2119894119898 (x 119905) = 0 on 120597Ω times [0 119905]

(41)

Here the parameters are V = 01 119863 = 001 1205821= 1205822= 01

and 119892 = 001We next treat the semidiscretised equation given by the

matrices

120597119905C = (

119860 minus Λ1minus 119866 0 119866 0

Λ1

119860 minus Λ2minus 119866 0 119866

119866 0 minusΛ1minus 119866 0

0 119866 Λ1

minusΛ2minus 119866

)C

(42)

where C = (c1 c2 c1119894119898 c2119894119898)119879 while c1 = (119888

11 119888

1119868) is the

solution of the first species in themobile phase in each spatialdiscretisation point (119894 = 1 119868) and the same for the othersolution vectors

We have the following two operators for the splittingmethod

119860 =119863

Δ1199092sdot (

minus2 1

1 minus2 1

d d d1 minus2 1

1 minus2

)

+VΔ119909

sdot (

1

minus1 1

d dminus1 1

minus1 1

) isin R119868times119868

(43)

where 119868 is the number of spatial pointsConsider

Λ1= (

1205821

0

0 1205821

0

d d d0 1205821

0

0 1205821

)

Λ2= (

1205822

0

0 1205822

0

d d d0 1205822

0

0 1205822

) isin R119868times119868

119866 = (

119892 0

0 119892 0

d d d0 119892 0

0 119892

) isin R119868times119868

(44)

We decouple this into the following matrices

119860 = (

119860 0 0 0

0 119860 0 0

0 0 0 0

0 0 0 0

) isin R4119868times4119868

1198611= (

minusΛ1

0 0 0

Λ1

minusΛ2

0 0

0 0 minusΛ1

0

0 0 Λ1

minusΛ2

)

1198612= (

minus119866 0 119866 0

0 minus119866 0 119866

119866 0 minus119866 0

0 119866 0 minus119866

) isin R4119868times4119868

(45)

For the operators 119860 and 119861 = 1198611+ 1198612and 119861 we apply the

iterative splitting method given in Section 21Based on the decomposition119860 is block diagonal and 119861 is

tridiagonalThe commutator is

[119860 119861] = (

0 0 119860119866 0

0 0 0 119860119866

minus119860119866 0 0 0

0 minus119860119866 0 0

) isin R4119868times4119868

(46)

10 International Journal of Differential Equations

Strang

AB

102

101

100

10minus1

10minus2

10minus3

100

Δt

c1

c2

c3

c4

c5

c6

Erro

r L1

AB Strang one-sided with B

(a)

CPU

tim

e

AB Strang one-side

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

(b)

Figure 5 Numerical errors and CPU time of the standard splittingscheme and the iterative schemes with Zassenhaus formula wherewe have at least 1 4 iterative steps

where we have a sparser operator than the main operator 119861and therefore we save computation time in computing suchoperators Further we assume that we have a bounded spatialstep size which is given by Δ119909 = 01 ≫ 0 so that we scale thesingular entry119863Δ119909

2asymp 1 and VΔ119909 asymp 1

Figure 5 presents the numerical error that is the differ-ence between the exact and the numerical solution and theCPU time required The optimal results are from the one-sided iterative schemes on the operator 119861

Remark 16 We obtain the same results as in the one-phase example because of the basically constant CPUtime for decreasing time steps of the iterative scheme weobtain optimal results when applying 1ndash3 Zassenhaus stepsTherefore with the embedded Zassenhaus formula we can

reach improved results more quickly than with the standardschemes With 2-3 more iterative steps we obtain moreaccurate results With the one-sided iterative schemes wereach the best convergence results when using the moredelicate operator 119861

4 Conclusions and Discussion

In this paper we have presented a novel splitting schemecombining the ideas of iterative and sequential schemes basedon the Zassenhaus formula Here the idea is to decouplethe expensive iterative computation of the schemes basedonly on the matrix exponential obtaining simpler embeddedZassenhaus schemes based on nilpotent commutators Suchcombinations have the benefit of reducing the computationtime because the commutators can be computed very cheaplyOn the other hand simple linear iterative steps can be donevery cheaply if they are initialised with a sufficiently accuratestarting solution and this accelerates the solvers The erroranalysis showed that these are stablemethods for higher orderschemes In the applications we were able to demonstratethe speedup from the Zassenhaus enhanced method andwere able to apply it to multiphysics applications In thefuture we will concentrate on linear and nonlinear matrix-dependent schemes that switch between noniterative anditerative schemes based on Zassenhaus products

References

[1] R E Ewing ldquoUp-scaling of biological processes andmultiphaseow in porous mediardquo in IIMA Volumes in Mathematics andIts Applications vol 295 pp 195ndash215 Springer New York NYUSA 2002

[2] J Geiser Discretization and Simulation of Systems forConvection-Diffusion-Dispersion Reactions with Applicationsin Groundwater Contamination Groundwater ModellingManagement and Contamination Nova Science MonographNew York NY USA 2008

[3] N Antonic C J van Duijn W Jager and A MikelicMultiscaleProblems in Science and Technology Challenges toMathematicalAnalysis and Perspectives Springer Berlin Germany 2002

[4] E Weinan Principle of Multiscale Modelling Cambridge Uni-versity Press Cambridge UK 2011

[5] S Blanes and F Casas ldquoOn the necessity of negative coefficientsfor operator splitting schemes of order higher than twordquoAppliedNumerical Mathematics vol 54 no 1 pp 23ndash37 2005

[6] EHansen andAOstermann ldquoHigh order splittingmethods foranalytic semigroups existrdquo BIT NumericalMathematics vol 49no 3 pp 527ndash542 2009

[7] G Strang ldquoOn the construction and comparison of differenceschemesrdquo SIAM Journal on Numerical Analysis vol 5 pp 506ndash517 1968

[8] G I Marchuk ldquoSome applicatons of splitting-up methods tothe solution of problems in mathematical physicsrdquo AplikaceMatematiky vol 1 pp 103ndash132 1968

[9] J Geiser ldquoComputing exponential for iterative splitting meth-ods algorithms and applicationsrdquo Journal of AppliedMathemat-ics vol 2011 Article ID 193781 27 pages 2011

[10] R I McLachlan and G RW Quispel ldquoSplittingmethodsrdquoActaNumerica vol 11 pp 341ndash434 2002

International Journal of Differential Equations 11

[11] J Geiser G Tanoglu and N Gucuyenen ldquoHigher order oper-ator splitting methods via Zassenhaus product formula theoryand applicationsrdquo Computers amp Mathematics with Applicationsvol 62 no 4 pp 1994ndash2015 2011

[12] G H Weiss and A A Maradudin ldquoThe Baker-Hausdorff for-mula and a problem in crystal physicsrdquo Journal of MathematicalPhysics vol 3 pp 771ndash777 1962

[13] R MWilcox ldquoExponential operators and parameter differenti-ation in quantum physicsrdquo Journal of Mathematical Physics vol8 pp 962ndash982 1967

[14] J Geiser ldquoAn iterative splitting approach for linear integro-differential equationsrdquo Applied Mathematics Letters vol 26 no11 pp 1048ndash1052 2013

[15] F Casas and A Murua ldquoAn efficient algorithm for computingthe Baker-Campbell-Hausdorff series and some of its applica-tionsrdquo Journal of Mathematical Physics vol 50 no 3 Article ID033513 2009

[16] J Geiser Iterative Splitting Methods for Differential EquationsChapmanampHallCRCNumerical Analysis and Scientific Com-puting CRC Press Boca Raton Fla USA 2011

[17] S Vandewalle Parallel Multigrid Waveform Relaxation forParabolic Problems Teubner Stuttgart Germany 1993

[18] J Geiser and G Tanoglu ldquoOperator-splitting methods viathe Zassenhaus product formulardquo Applied Mathematics andComputation vol 217 no 9 pp 4557ndash4575 2011

[19] F Bayen ldquoOn the convergence of the Zassenhaus formulardquoLetters in Mathematical Physics vol 3 no 3 pp 161ndash167 1979

[20] F Casas A Murua and M Nadinic ldquoEfficient computation ofthe Zassenhaus formulardquo Computer Physics Communicationsvol 183 no 11 pp 2386ndash2391 2012

[21] E Hairer C Lubich and GWannerGeometric Numerical Inte-gration vol 31 of Springer Series in Computational MathematicsSpringer Berlin Germany 2002

[22] H Yoshida ldquoConstruction of higher order symplectic integra-torsrdquo Physics Letters A vol 150 no 5ndash7 pp 262ndash268 1990

[23] E Faou A Ostermann and K Schratz ldquoAnalysis of exponentialsplitting methods for inhomogeneous parabolic equationsrdquohttparxivorgabs12125827

[24] J Geiser ldquoDiscretization methods with embedded analyticalsolutions for convection-diffusion dispersion-reaction equa-tions and applicationsrdquo Journal of EngineeringMathematics vol57 no 1 pp 79ndash98 2007

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 International Journal of Differential Equations

102

102

10minus2

10minus4

10minus6

10minus8

10minus2

10minus1

100

AB Strang one-sided

AB

Δt

Strang

c1

c2

c3

c4

c5

c6

Erro

r L1

(a)

AB

Δt

Strang

102

102

10minus2

10minus4

10minus6

10minus8

10minus2

10minus1

100

AB Strang two-sided

c1

c2

c3

c4

c5

c6

Erro

r L1

(b)

Figure 1 Numerical errors of the standard splitting scheme and theiterative schemes with the Zassenhaus formula where we have atleast 1 4 iterative steps

Remark 13 We see that the errors decrease significantlywith increasing order of approximation for the initialisa-tion process from the Zassenhaus splitting (we apply 1ndash3Zassenhaus coefficients) Here we apply commutators whichdid not reduce their matrix rank therefore we have thesame computation time as for the pure iterative schemesOverall we have their benefits in the application of theZassenhaus product schemes to the standard Lie-Trotter orStrang-Marchuk splitting Similar results are given withmoreiterative steps 119894 = 3 6 as expected

32 Real Life Problems Multiphase Examples In the nextexamples we simulate coupled transport and reaction equa-tions with different phases so-called multiphase problems

CPU

tim

e

AB Strang one-sided

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

(a)

CPU

tim

e

AB Strang two-sided

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

(b)

Figure 2 CPU times of the standard splitting scheme and theiterative schemes with the Zassenhaus formula where we have atleast 1 4 iterative steps

see [9]The equations are coupledwith the reaction terms andare presented as follows

120597119905119877119894119906119894+ nabla sdot v119906

119894= minus 120582

119894119877119894119906119894+ 120582119894minus1

119877119894minus1

119906119894minus1

+ 120573 (minus119906119894+ 119892119894) inΩ times (0 119879)

1199061198940

(119909) = 119906119894(119909 0) on Ω

120597119905119877119894119892119894= minus 120582

119894119877119894119892119894+ 120582119894minus1

119877119894minus1

119892119894minus1

+ 120573 (minus119892119894+ 119906119894) in Ω times (0 119879)

1198921198940

(119909) = 119892119894(119909 0) on Ω

119894 = 1 119898

(34)

International Journal of Differential Equations 7

where 119898 is the number of equations and 119894 is the index ofeach component The unknown mobile concentrations 119906

119894=

119906119894(119909 119905) are taken in Ω times (0 119879) sub R119899 times R+ where 119899 is the

spatial dimension The unknown immobile concentrations119892119894= 119892119894(119909 119905) are taken in Ω times (0 119879) sub R119899 times R+ where 119899 is

the spatial dimensionThe retardation factors119877119894are constant

and 119877119894

ge 0 The kinetic part is given by the factors 120582119894

They are constant and 120582119894ge 0 For the initialisation of the

kinetic part we set 1205820

= 0 The kinetic part is linear andirreversible so the successors have only one predecessor Theinitial conditions are given for each component 119894 as constantsor linear impulses For the boundary conditions we havetrivial inflow and outflow conditions with 119906

119894= 0 at the inflow

boundary The transport part is given by the velocity v isin R119899

and is piecewise constant see [2] The exchange between themobile and immobile part is given by 120573

In the following we discuss the one-phase and two-phaseexamples

321 One-Phase Example The next example is a simplifiedreal life problem with a multiphase transport-reaction equa-tion We treat the mobile and immobile pores in a porousmedia Such simulations are made about waste scenarios see[2 24]

We concentrate on the computational benefits of a fastcomputation of the mixed iterative scheme with the Zassen-haus formula

The one phase equation is

1205971199051198881+ nabla sdot F

11198881= minus12058211198881 in Ω times [0 119905]

1205971199051198882+ nabla sdot F

21198882= 12058211198881minus 12058221198882 in Ω times [0 119905]

F119894= v119894minus 119863119894nabla 119894 = 1 2

1198881(x 119905) = 119888

10(x) 119888

2(x 119905) = 119888

20(x) on Ω

1198881 (x 119905) = 119888

11 (x 119905) 1198882 (x 119905) = 119888

21 (x 119905)

on 120597Ω times [0 119905]

(35)

Here the parameters take the values V1= 01 V

2= 005119863

1=

0011198632= 0005 and 120582

1= 1205822= 01

We now turn to the finite difference schemes andsemidiscretise the equation with its convection and diffusionoperators as follows

120597119905c = (119860

1+ 1198602) c (36)

We obtain the two matrices and decouple the diffusion andconvection parts as follows

1198601= (

119860diff 0

0 119860diff) isin R

2119868times2119868

1198602= (

119860Conv 0

0 119860Conv) + (

minusΛ1

0

Λ1

minusΛ2

) isin R2119868times2119868

(37)

We apply the splitting method to the operators 1198601and 119860

2

given in Section 21

The submatrices are

119860diff119894 =119863119894

Δ11990921198771=

119863119894

Δ1199092sdot (

minus2 1

1 minus2 1

d d d1 minus2 1

1 minus2

) isin R119868times119868

119860conv119894 = minusV119894

Δ1199091198772= minus

V119894

Δ119909sdot (

1

minus1 1

d dminus1 1

minus1 1

) isin R119868times119868

(38)

where 119868 is the number of spatial points and Δ119909 is the spatialstep size Consider

Λ1= (

1205821

0

0 1205821

0

d d d0 1205821

0

0 1205821

) isin R119868times119868

Λ2= (

1205822

0

0 1205822

0

d d d0 1205822

0

0 1205822

) isin R119868times119868

(39)

We have the commutator

[1198601 1198602] = (

0 0

1198632minus 1198631

Δ11990921198771Λ1

0) isin R

2119868times2119868 (40)

where we assume that [1198771 1198772] asymp 0 This is a nilpotent

commutator and the computation time needed for thecommutators of such operators is less than that for the fulloperators Furthermore we assume that we have a boundedspatial step size which is given by Δ119909 = 01 ≫ 0 so thatwe scale the singular entry (119863

2minus 1198631)Δ1199092asymp 1 and evade a

blowup in such matricesWe have obtained the optimal results of a basically

constant CPU time for decreasing time steps which increasein the standard scheme

We apply 10ndash100 timesteps for the computations with 10ndash100 spatial points The final integration time of each singletime step was about 1ndash10 [sec] such that we need at least forsuch benchmark problems about 100ndash1000 [sec]

Figure 3 presents the numerical error that is the differ-ence between the exact and the numerical solution

Figure 4 presents the CPU time of the standard andthe iterative splitting schemes Based on the preprocess ofcomputing the Zassenhaus exponents at the beginning andstore the matrices we could save additional computationaltime with larger time steps

Remark 14 For the iterative schemes with embedded Zassen-haus products we can reach improved results more quicklyThe benefits come from the constant CPU time needed for

8 International Journal of Differential Equations

AB

Δt

Strang

101

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

Erro

r L1

AB Strang one-sided with A

(a)

AB

Δt

Strang

102

100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

Erro

r L1

AB Strang one-sided with B

(b)

AB

Δt

Strang

102

100

10minus2

10minus4

10minus6

10minus8

10minus2

10minus1

100

AB Strang two-side

c1

c2

c3

c4

c5

c6

Erro

r L1

(c)

Figure 3 Numerical errors of the standard splitting scheme and theiterative schemes with the Zassenhaus formula where we have atleast 1 4 iterative steps

CPU

tim

e

AB strang one-side with A

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

(a)

CPU

tim

e

Δt

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

AB

Strang

c1

c2

c3

c4

c5

c6

AB Strang one-sided with B

(b)

CPU

tim

e

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

AB Strang two-side

c1

c2

c3

c4

c5

c6

(c)

Figure 4 CPU time of the standard splitting scheme and theiterative schemes with the Zassenhaus formula where we have atleast 1 4 iterative steps

International Journal of Differential Equations 9

calculating the nilpotent commutators in the Zassenhausformula So it makes sense to have 1ndash3 Zassenhaus stepsbefore starting the iterative scheme At that point only 2-3iterative steps are needed to obtainmore accurate results thanthose from the expensive standard schemes Here too weobtain the best results with the one-sided iterative schemes

Remark 15 Based on our restriction to 1198631 1198632 and Δ119909 with

(1198632minus 1198631)Δ1199092

asymp 1 we can control the delicate blowupof the computed commutators By the way if we neglectthe restriction and Δ119909 with (119863

2minus 1198631)Δ1199092

rarr infin thecommutator blows up and we lose the benefits of such analgorithmThis is a limitation of applying such an embeddedZassenhaus formula

322 Two Phase Example The next example is a moredelicate real life problem involving a multiphase transport-reaction equation We treat mobile and immobile pores in aporousmedia Such simulations aremade forwaste scenariossee [2]

We concentrate on the computational benefits of a fastcomputation of the commutators and their use for theinitialisation of the iterative splitting scheme

The equation is

1205971199051198881+ nabla sdot F119888

1= 119892 (minus119888

1+ 1198881119894119898

) minus 12058211198881 in Ω times [0 119905]

1205971199051198882+ nabla sdot F119888

2= 119892 (minus119888

2+ 1198882119894119898

) + 12058211198881minus 12058221198882 in Ω times [0 119905]

F = v minus 119863nabla

1205971199051198881119894119898

= 119892 (1198881minus 1198881119894119898

) minus 12058211198881119894119898

in Ω times [0 119905]

1205971199051198882119894119898

= 119892 (1198882minus 1198882119894119898

) + 12058211198881119894119898

minus 12058221198882119894119898

in Ω times [0 119905]

1198881(x 119905) = 119888

10(x) 119888

2(x 119905) = 119888

20(x) on Ω

1198881(x 119905) = 119888

11(x 119905) 119888

2(x 119905) = 119888

21(x 119905)

on 120597Ω times [0 119905]

1198881119894119898

(x 119905) = 0 1198882119894119898

(x 119905) = 0 on Ω

1198881119894119898 (x 119905) = 0 119888

2119894119898 (x 119905) = 0 on 120597Ω times [0 119905]

(41)

Here the parameters are V = 01 119863 = 001 1205821= 1205822= 01

and 119892 = 001We next treat the semidiscretised equation given by the

matrices

120597119905C = (

119860 minus Λ1minus 119866 0 119866 0

Λ1

119860 minus Λ2minus 119866 0 119866

119866 0 minusΛ1minus 119866 0

0 119866 Λ1

minusΛ2minus 119866

)C

(42)

where C = (c1 c2 c1119894119898 c2119894119898)119879 while c1 = (119888

11 119888

1119868) is the

solution of the first species in themobile phase in each spatialdiscretisation point (119894 = 1 119868) and the same for the othersolution vectors

We have the following two operators for the splittingmethod

119860 =119863

Δ1199092sdot (

minus2 1

1 minus2 1

d d d1 minus2 1

1 minus2

)

+VΔ119909

sdot (

1

minus1 1

d dminus1 1

minus1 1

) isin R119868times119868

(43)

where 119868 is the number of spatial pointsConsider

Λ1= (

1205821

0

0 1205821

0

d d d0 1205821

0

0 1205821

)

Λ2= (

1205822

0

0 1205822

0

d d d0 1205822

0

0 1205822

) isin R119868times119868

119866 = (

119892 0

0 119892 0

d d d0 119892 0

0 119892

) isin R119868times119868

(44)

We decouple this into the following matrices

119860 = (

119860 0 0 0

0 119860 0 0

0 0 0 0

0 0 0 0

) isin R4119868times4119868

1198611= (

minusΛ1

0 0 0

Λ1

minusΛ2

0 0

0 0 minusΛ1

0

0 0 Λ1

minusΛ2

)

1198612= (

minus119866 0 119866 0

0 minus119866 0 119866

119866 0 minus119866 0

0 119866 0 minus119866

) isin R4119868times4119868

(45)

For the operators 119860 and 119861 = 1198611+ 1198612and 119861 we apply the

iterative splitting method given in Section 21Based on the decomposition119860 is block diagonal and 119861 is

tridiagonalThe commutator is

[119860 119861] = (

0 0 119860119866 0

0 0 0 119860119866

minus119860119866 0 0 0

0 minus119860119866 0 0

) isin R4119868times4119868

(46)

10 International Journal of Differential Equations

Strang

AB

102

101

100

10minus1

10minus2

10minus3

100

Δt

c1

c2

c3

c4

c5

c6

Erro

r L1

AB Strang one-sided with B

(a)

CPU

tim

e

AB Strang one-side

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

(b)

Figure 5 Numerical errors and CPU time of the standard splittingscheme and the iterative schemes with Zassenhaus formula wherewe have at least 1 4 iterative steps

where we have a sparser operator than the main operator 119861and therefore we save computation time in computing suchoperators Further we assume that we have a bounded spatialstep size which is given by Δ119909 = 01 ≫ 0 so that we scale thesingular entry119863Δ119909

2asymp 1 and VΔ119909 asymp 1

Figure 5 presents the numerical error that is the differ-ence between the exact and the numerical solution and theCPU time required The optimal results are from the one-sided iterative schemes on the operator 119861

Remark 16 We obtain the same results as in the one-phase example because of the basically constant CPUtime for decreasing time steps of the iterative scheme weobtain optimal results when applying 1ndash3 Zassenhaus stepsTherefore with the embedded Zassenhaus formula we can

reach improved results more quickly than with the standardschemes With 2-3 more iterative steps we obtain moreaccurate results With the one-sided iterative schemes wereach the best convergence results when using the moredelicate operator 119861

4 Conclusions and Discussion

In this paper we have presented a novel splitting schemecombining the ideas of iterative and sequential schemes basedon the Zassenhaus formula Here the idea is to decouplethe expensive iterative computation of the schemes basedonly on the matrix exponential obtaining simpler embeddedZassenhaus schemes based on nilpotent commutators Suchcombinations have the benefit of reducing the computationtime because the commutators can be computed very cheaplyOn the other hand simple linear iterative steps can be donevery cheaply if they are initialised with a sufficiently accuratestarting solution and this accelerates the solvers The erroranalysis showed that these are stablemethods for higher orderschemes In the applications we were able to demonstratethe speedup from the Zassenhaus enhanced method andwere able to apply it to multiphysics applications In thefuture we will concentrate on linear and nonlinear matrix-dependent schemes that switch between noniterative anditerative schemes based on Zassenhaus products

References

[1] R E Ewing ldquoUp-scaling of biological processes andmultiphaseow in porous mediardquo in IIMA Volumes in Mathematics andIts Applications vol 295 pp 195ndash215 Springer New York NYUSA 2002

[2] J Geiser Discretization and Simulation of Systems forConvection-Diffusion-Dispersion Reactions with Applicationsin Groundwater Contamination Groundwater ModellingManagement and Contamination Nova Science MonographNew York NY USA 2008

[3] N Antonic C J van Duijn W Jager and A MikelicMultiscaleProblems in Science and Technology Challenges toMathematicalAnalysis and Perspectives Springer Berlin Germany 2002

[4] E Weinan Principle of Multiscale Modelling Cambridge Uni-versity Press Cambridge UK 2011

[5] S Blanes and F Casas ldquoOn the necessity of negative coefficientsfor operator splitting schemes of order higher than twordquoAppliedNumerical Mathematics vol 54 no 1 pp 23ndash37 2005

[6] EHansen andAOstermann ldquoHigh order splittingmethods foranalytic semigroups existrdquo BIT NumericalMathematics vol 49no 3 pp 527ndash542 2009

[7] G Strang ldquoOn the construction and comparison of differenceschemesrdquo SIAM Journal on Numerical Analysis vol 5 pp 506ndash517 1968

[8] G I Marchuk ldquoSome applicatons of splitting-up methods tothe solution of problems in mathematical physicsrdquo AplikaceMatematiky vol 1 pp 103ndash132 1968

[9] J Geiser ldquoComputing exponential for iterative splitting meth-ods algorithms and applicationsrdquo Journal of AppliedMathemat-ics vol 2011 Article ID 193781 27 pages 2011

[10] R I McLachlan and G RW Quispel ldquoSplittingmethodsrdquoActaNumerica vol 11 pp 341ndash434 2002

International Journal of Differential Equations 11

[11] J Geiser G Tanoglu and N Gucuyenen ldquoHigher order oper-ator splitting methods via Zassenhaus product formula theoryand applicationsrdquo Computers amp Mathematics with Applicationsvol 62 no 4 pp 1994ndash2015 2011

[12] G H Weiss and A A Maradudin ldquoThe Baker-Hausdorff for-mula and a problem in crystal physicsrdquo Journal of MathematicalPhysics vol 3 pp 771ndash777 1962

[13] R MWilcox ldquoExponential operators and parameter differenti-ation in quantum physicsrdquo Journal of Mathematical Physics vol8 pp 962ndash982 1967

[14] J Geiser ldquoAn iterative splitting approach for linear integro-differential equationsrdquo Applied Mathematics Letters vol 26 no11 pp 1048ndash1052 2013

[15] F Casas and A Murua ldquoAn efficient algorithm for computingthe Baker-Campbell-Hausdorff series and some of its applica-tionsrdquo Journal of Mathematical Physics vol 50 no 3 Article ID033513 2009

[16] J Geiser Iterative Splitting Methods for Differential EquationsChapmanampHallCRCNumerical Analysis and Scientific Com-puting CRC Press Boca Raton Fla USA 2011

[17] S Vandewalle Parallel Multigrid Waveform Relaxation forParabolic Problems Teubner Stuttgart Germany 1993

[18] J Geiser and G Tanoglu ldquoOperator-splitting methods viathe Zassenhaus product formulardquo Applied Mathematics andComputation vol 217 no 9 pp 4557ndash4575 2011

[19] F Bayen ldquoOn the convergence of the Zassenhaus formulardquoLetters in Mathematical Physics vol 3 no 3 pp 161ndash167 1979

[20] F Casas A Murua and M Nadinic ldquoEfficient computation ofthe Zassenhaus formulardquo Computer Physics Communicationsvol 183 no 11 pp 2386ndash2391 2012

[21] E Hairer C Lubich and GWannerGeometric Numerical Inte-gration vol 31 of Springer Series in Computational MathematicsSpringer Berlin Germany 2002

[22] H Yoshida ldquoConstruction of higher order symplectic integra-torsrdquo Physics Letters A vol 150 no 5ndash7 pp 262ndash268 1990

[23] E Faou A Ostermann and K Schratz ldquoAnalysis of exponentialsplitting methods for inhomogeneous parabolic equationsrdquohttparxivorgabs12125827

[24] J Geiser ldquoDiscretization methods with embedded analyticalsolutions for convection-diffusion dispersion-reaction equa-tions and applicationsrdquo Journal of EngineeringMathematics vol57 no 1 pp 79ndash98 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Differential Equations 7

where 119898 is the number of equations and 119894 is the index ofeach component The unknown mobile concentrations 119906

119894=

119906119894(119909 119905) are taken in Ω times (0 119879) sub R119899 times R+ where 119899 is the

spatial dimension The unknown immobile concentrations119892119894= 119892119894(119909 119905) are taken in Ω times (0 119879) sub R119899 times R+ where 119899 is

the spatial dimensionThe retardation factors119877119894are constant

and 119877119894

ge 0 The kinetic part is given by the factors 120582119894

They are constant and 120582119894ge 0 For the initialisation of the

kinetic part we set 1205820

= 0 The kinetic part is linear andirreversible so the successors have only one predecessor Theinitial conditions are given for each component 119894 as constantsor linear impulses For the boundary conditions we havetrivial inflow and outflow conditions with 119906

119894= 0 at the inflow

boundary The transport part is given by the velocity v isin R119899

and is piecewise constant see [2] The exchange between themobile and immobile part is given by 120573

In the following we discuss the one-phase and two-phaseexamples

321 One-Phase Example The next example is a simplifiedreal life problem with a multiphase transport-reaction equa-tion We treat the mobile and immobile pores in a porousmedia Such simulations are made about waste scenarios see[2 24]

We concentrate on the computational benefits of a fastcomputation of the mixed iterative scheme with the Zassen-haus formula

The one phase equation is

1205971199051198881+ nabla sdot F

11198881= minus12058211198881 in Ω times [0 119905]

1205971199051198882+ nabla sdot F

21198882= 12058211198881minus 12058221198882 in Ω times [0 119905]

F119894= v119894minus 119863119894nabla 119894 = 1 2

1198881(x 119905) = 119888

10(x) 119888

2(x 119905) = 119888

20(x) on Ω

1198881 (x 119905) = 119888

11 (x 119905) 1198882 (x 119905) = 119888

21 (x 119905)

on 120597Ω times [0 119905]

(35)

Here the parameters take the values V1= 01 V

2= 005119863

1=

0011198632= 0005 and 120582

1= 1205822= 01

We now turn to the finite difference schemes andsemidiscretise the equation with its convection and diffusionoperators as follows

120597119905c = (119860

1+ 1198602) c (36)

We obtain the two matrices and decouple the diffusion andconvection parts as follows

1198601= (

119860diff 0

0 119860diff) isin R

2119868times2119868

1198602= (

119860Conv 0

0 119860Conv) + (

minusΛ1

0

Λ1

minusΛ2

) isin R2119868times2119868

(37)

We apply the splitting method to the operators 1198601and 119860

2

given in Section 21

The submatrices are

119860diff119894 =119863119894

Δ11990921198771=

119863119894

Δ1199092sdot (

minus2 1

1 minus2 1

d d d1 minus2 1

1 minus2

) isin R119868times119868

119860conv119894 = minusV119894

Δ1199091198772= minus

V119894

Δ119909sdot (

1

minus1 1

d dminus1 1

minus1 1

) isin R119868times119868

(38)

where 119868 is the number of spatial points and Δ119909 is the spatialstep size Consider

Λ1= (

1205821

0

0 1205821

0

d d d0 1205821

0

0 1205821

) isin R119868times119868

Λ2= (

1205822

0

0 1205822

0

d d d0 1205822

0

0 1205822

) isin R119868times119868

(39)

We have the commutator

[1198601 1198602] = (

0 0

1198632minus 1198631

Δ11990921198771Λ1

0) isin R

2119868times2119868 (40)

where we assume that [1198771 1198772] asymp 0 This is a nilpotent

commutator and the computation time needed for thecommutators of such operators is less than that for the fulloperators Furthermore we assume that we have a boundedspatial step size which is given by Δ119909 = 01 ≫ 0 so thatwe scale the singular entry (119863

2minus 1198631)Δ1199092asymp 1 and evade a

blowup in such matricesWe have obtained the optimal results of a basically

constant CPU time for decreasing time steps which increasein the standard scheme

We apply 10ndash100 timesteps for the computations with 10ndash100 spatial points The final integration time of each singletime step was about 1ndash10 [sec] such that we need at least forsuch benchmark problems about 100ndash1000 [sec]

Figure 3 presents the numerical error that is the differ-ence between the exact and the numerical solution

Figure 4 presents the CPU time of the standard andthe iterative splitting schemes Based on the preprocess ofcomputing the Zassenhaus exponents at the beginning andstore the matrices we could save additional computationaltime with larger time steps

Remark 14 For the iterative schemes with embedded Zassen-haus products we can reach improved results more quicklyThe benefits come from the constant CPU time needed for

8 International Journal of Differential Equations

AB

Δt

Strang

101

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

Erro

r L1

AB Strang one-sided with A

(a)

AB

Δt

Strang

102

100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

Erro

r L1

AB Strang one-sided with B

(b)

AB

Δt

Strang

102

100

10minus2

10minus4

10minus6

10minus8

10minus2

10minus1

100

AB Strang two-side

c1

c2

c3

c4

c5

c6

Erro

r L1

(c)

Figure 3 Numerical errors of the standard splitting scheme and theiterative schemes with the Zassenhaus formula where we have atleast 1 4 iterative steps

CPU

tim

e

AB strang one-side with A

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

(a)

CPU

tim

e

Δt

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

AB

Strang

c1

c2

c3

c4

c5

c6

AB Strang one-sided with B

(b)

CPU

tim

e

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

AB Strang two-side

c1

c2

c3

c4

c5

c6

(c)

Figure 4 CPU time of the standard splitting scheme and theiterative schemes with the Zassenhaus formula where we have atleast 1 4 iterative steps

International Journal of Differential Equations 9

calculating the nilpotent commutators in the Zassenhausformula So it makes sense to have 1ndash3 Zassenhaus stepsbefore starting the iterative scheme At that point only 2-3iterative steps are needed to obtainmore accurate results thanthose from the expensive standard schemes Here too weobtain the best results with the one-sided iterative schemes

Remark 15 Based on our restriction to 1198631 1198632 and Δ119909 with

(1198632minus 1198631)Δ1199092

asymp 1 we can control the delicate blowupof the computed commutators By the way if we neglectthe restriction and Δ119909 with (119863

2minus 1198631)Δ1199092

rarr infin thecommutator blows up and we lose the benefits of such analgorithmThis is a limitation of applying such an embeddedZassenhaus formula

322 Two Phase Example The next example is a moredelicate real life problem involving a multiphase transport-reaction equation We treat mobile and immobile pores in aporousmedia Such simulations aremade forwaste scenariossee [2]

We concentrate on the computational benefits of a fastcomputation of the commutators and their use for theinitialisation of the iterative splitting scheme

The equation is

1205971199051198881+ nabla sdot F119888

1= 119892 (minus119888

1+ 1198881119894119898

) minus 12058211198881 in Ω times [0 119905]

1205971199051198882+ nabla sdot F119888

2= 119892 (minus119888

2+ 1198882119894119898

) + 12058211198881minus 12058221198882 in Ω times [0 119905]

F = v minus 119863nabla

1205971199051198881119894119898

= 119892 (1198881minus 1198881119894119898

) minus 12058211198881119894119898

in Ω times [0 119905]

1205971199051198882119894119898

= 119892 (1198882minus 1198882119894119898

) + 12058211198881119894119898

minus 12058221198882119894119898

in Ω times [0 119905]

1198881(x 119905) = 119888

10(x) 119888

2(x 119905) = 119888

20(x) on Ω

1198881(x 119905) = 119888

11(x 119905) 119888

2(x 119905) = 119888

21(x 119905)

on 120597Ω times [0 119905]

1198881119894119898

(x 119905) = 0 1198882119894119898

(x 119905) = 0 on Ω

1198881119894119898 (x 119905) = 0 119888

2119894119898 (x 119905) = 0 on 120597Ω times [0 119905]

(41)

Here the parameters are V = 01 119863 = 001 1205821= 1205822= 01

and 119892 = 001We next treat the semidiscretised equation given by the

matrices

120597119905C = (

119860 minus Λ1minus 119866 0 119866 0

Λ1

119860 minus Λ2minus 119866 0 119866

119866 0 minusΛ1minus 119866 0

0 119866 Λ1

minusΛ2minus 119866

)C

(42)

where C = (c1 c2 c1119894119898 c2119894119898)119879 while c1 = (119888

11 119888

1119868) is the

solution of the first species in themobile phase in each spatialdiscretisation point (119894 = 1 119868) and the same for the othersolution vectors

We have the following two operators for the splittingmethod

119860 =119863

Δ1199092sdot (

minus2 1

1 minus2 1

d d d1 minus2 1

1 minus2

)

+VΔ119909

sdot (

1

minus1 1

d dminus1 1

minus1 1

) isin R119868times119868

(43)

where 119868 is the number of spatial pointsConsider

Λ1= (

1205821

0

0 1205821

0

d d d0 1205821

0

0 1205821

)

Λ2= (

1205822

0

0 1205822

0

d d d0 1205822

0

0 1205822

) isin R119868times119868

119866 = (

119892 0

0 119892 0

d d d0 119892 0

0 119892

) isin R119868times119868

(44)

We decouple this into the following matrices

119860 = (

119860 0 0 0

0 119860 0 0

0 0 0 0

0 0 0 0

) isin R4119868times4119868

1198611= (

minusΛ1

0 0 0

Λ1

minusΛ2

0 0

0 0 minusΛ1

0

0 0 Λ1

minusΛ2

)

1198612= (

minus119866 0 119866 0

0 minus119866 0 119866

119866 0 minus119866 0

0 119866 0 minus119866

) isin R4119868times4119868

(45)

For the operators 119860 and 119861 = 1198611+ 1198612and 119861 we apply the

iterative splitting method given in Section 21Based on the decomposition119860 is block diagonal and 119861 is

tridiagonalThe commutator is

[119860 119861] = (

0 0 119860119866 0

0 0 0 119860119866

minus119860119866 0 0 0

0 minus119860119866 0 0

) isin R4119868times4119868

(46)

10 International Journal of Differential Equations

Strang

AB

102

101

100

10minus1

10minus2

10minus3

100

Δt

c1

c2

c3

c4

c5

c6

Erro

r L1

AB Strang one-sided with B

(a)

CPU

tim

e

AB Strang one-side

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

(b)

Figure 5 Numerical errors and CPU time of the standard splittingscheme and the iterative schemes with Zassenhaus formula wherewe have at least 1 4 iterative steps

where we have a sparser operator than the main operator 119861and therefore we save computation time in computing suchoperators Further we assume that we have a bounded spatialstep size which is given by Δ119909 = 01 ≫ 0 so that we scale thesingular entry119863Δ119909

2asymp 1 and VΔ119909 asymp 1

Figure 5 presents the numerical error that is the differ-ence between the exact and the numerical solution and theCPU time required The optimal results are from the one-sided iterative schemes on the operator 119861

Remark 16 We obtain the same results as in the one-phase example because of the basically constant CPUtime for decreasing time steps of the iterative scheme weobtain optimal results when applying 1ndash3 Zassenhaus stepsTherefore with the embedded Zassenhaus formula we can

reach improved results more quickly than with the standardschemes With 2-3 more iterative steps we obtain moreaccurate results With the one-sided iterative schemes wereach the best convergence results when using the moredelicate operator 119861

4 Conclusions and Discussion

In this paper we have presented a novel splitting schemecombining the ideas of iterative and sequential schemes basedon the Zassenhaus formula Here the idea is to decouplethe expensive iterative computation of the schemes basedonly on the matrix exponential obtaining simpler embeddedZassenhaus schemes based on nilpotent commutators Suchcombinations have the benefit of reducing the computationtime because the commutators can be computed very cheaplyOn the other hand simple linear iterative steps can be donevery cheaply if they are initialised with a sufficiently accuratestarting solution and this accelerates the solvers The erroranalysis showed that these are stablemethods for higher orderschemes In the applications we were able to demonstratethe speedup from the Zassenhaus enhanced method andwere able to apply it to multiphysics applications In thefuture we will concentrate on linear and nonlinear matrix-dependent schemes that switch between noniterative anditerative schemes based on Zassenhaus products

References

[1] R E Ewing ldquoUp-scaling of biological processes andmultiphaseow in porous mediardquo in IIMA Volumes in Mathematics andIts Applications vol 295 pp 195ndash215 Springer New York NYUSA 2002

[2] J Geiser Discretization and Simulation of Systems forConvection-Diffusion-Dispersion Reactions with Applicationsin Groundwater Contamination Groundwater ModellingManagement and Contamination Nova Science MonographNew York NY USA 2008

[3] N Antonic C J van Duijn W Jager and A MikelicMultiscaleProblems in Science and Technology Challenges toMathematicalAnalysis and Perspectives Springer Berlin Germany 2002

[4] E Weinan Principle of Multiscale Modelling Cambridge Uni-versity Press Cambridge UK 2011

[5] S Blanes and F Casas ldquoOn the necessity of negative coefficientsfor operator splitting schemes of order higher than twordquoAppliedNumerical Mathematics vol 54 no 1 pp 23ndash37 2005

[6] EHansen andAOstermann ldquoHigh order splittingmethods foranalytic semigroups existrdquo BIT NumericalMathematics vol 49no 3 pp 527ndash542 2009

[7] G Strang ldquoOn the construction and comparison of differenceschemesrdquo SIAM Journal on Numerical Analysis vol 5 pp 506ndash517 1968

[8] G I Marchuk ldquoSome applicatons of splitting-up methods tothe solution of problems in mathematical physicsrdquo AplikaceMatematiky vol 1 pp 103ndash132 1968

[9] J Geiser ldquoComputing exponential for iterative splitting meth-ods algorithms and applicationsrdquo Journal of AppliedMathemat-ics vol 2011 Article ID 193781 27 pages 2011

[10] R I McLachlan and G RW Quispel ldquoSplittingmethodsrdquoActaNumerica vol 11 pp 341ndash434 2002

International Journal of Differential Equations 11

[11] J Geiser G Tanoglu and N Gucuyenen ldquoHigher order oper-ator splitting methods via Zassenhaus product formula theoryand applicationsrdquo Computers amp Mathematics with Applicationsvol 62 no 4 pp 1994ndash2015 2011

[12] G H Weiss and A A Maradudin ldquoThe Baker-Hausdorff for-mula and a problem in crystal physicsrdquo Journal of MathematicalPhysics vol 3 pp 771ndash777 1962

[13] R MWilcox ldquoExponential operators and parameter differenti-ation in quantum physicsrdquo Journal of Mathematical Physics vol8 pp 962ndash982 1967

[14] J Geiser ldquoAn iterative splitting approach for linear integro-differential equationsrdquo Applied Mathematics Letters vol 26 no11 pp 1048ndash1052 2013

[15] F Casas and A Murua ldquoAn efficient algorithm for computingthe Baker-Campbell-Hausdorff series and some of its applica-tionsrdquo Journal of Mathematical Physics vol 50 no 3 Article ID033513 2009

[16] J Geiser Iterative Splitting Methods for Differential EquationsChapmanampHallCRCNumerical Analysis and Scientific Com-puting CRC Press Boca Raton Fla USA 2011

[17] S Vandewalle Parallel Multigrid Waveform Relaxation forParabolic Problems Teubner Stuttgart Germany 1993

[18] J Geiser and G Tanoglu ldquoOperator-splitting methods viathe Zassenhaus product formulardquo Applied Mathematics andComputation vol 217 no 9 pp 4557ndash4575 2011

[19] F Bayen ldquoOn the convergence of the Zassenhaus formulardquoLetters in Mathematical Physics vol 3 no 3 pp 161ndash167 1979

[20] F Casas A Murua and M Nadinic ldquoEfficient computation ofthe Zassenhaus formulardquo Computer Physics Communicationsvol 183 no 11 pp 2386ndash2391 2012

[21] E Hairer C Lubich and GWannerGeometric Numerical Inte-gration vol 31 of Springer Series in Computational MathematicsSpringer Berlin Germany 2002

[22] H Yoshida ldquoConstruction of higher order symplectic integra-torsrdquo Physics Letters A vol 150 no 5ndash7 pp 262ndash268 1990

[23] E Faou A Ostermann and K Schratz ldquoAnalysis of exponentialsplitting methods for inhomogeneous parabolic equationsrdquohttparxivorgabs12125827

[24] J Geiser ldquoDiscretization methods with embedded analyticalsolutions for convection-diffusion dispersion-reaction equa-tions and applicationsrdquo Journal of EngineeringMathematics vol57 no 1 pp 79ndash98 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 International Journal of Differential Equations

AB

Δt

Strang

101

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

Erro

r L1

AB Strang one-sided with A

(a)

AB

Δt

Strang

102

100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

Erro

r L1

AB Strang one-sided with B

(b)

AB

Δt

Strang

102

100

10minus2

10minus4

10minus6

10minus8

10minus2

10minus1

100

AB Strang two-side

c1

c2

c3

c4

c5

c6

Erro

r L1

(c)

Figure 3 Numerical errors of the standard splitting scheme and theiterative schemes with the Zassenhaus formula where we have atleast 1 4 iterative steps

CPU

tim

e

AB strang one-side with A

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

(a)

CPU

tim

e

Δt

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

AB

Strang

c1

c2

c3

c4

c5

c6

AB Strang one-sided with B

(b)

CPU

tim

e

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

AB Strang two-side

c1

c2

c3

c4

c5

c6

(c)

Figure 4 CPU time of the standard splitting scheme and theiterative schemes with the Zassenhaus formula where we have atleast 1 4 iterative steps

International Journal of Differential Equations 9

calculating the nilpotent commutators in the Zassenhausformula So it makes sense to have 1ndash3 Zassenhaus stepsbefore starting the iterative scheme At that point only 2-3iterative steps are needed to obtainmore accurate results thanthose from the expensive standard schemes Here too weobtain the best results with the one-sided iterative schemes

Remark 15 Based on our restriction to 1198631 1198632 and Δ119909 with

(1198632minus 1198631)Δ1199092

asymp 1 we can control the delicate blowupof the computed commutators By the way if we neglectthe restriction and Δ119909 with (119863

2minus 1198631)Δ1199092

rarr infin thecommutator blows up and we lose the benefits of such analgorithmThis is a limitation of applying such an embeddedZassenhaus formula

322 Two Phase Example The next example is a moredelicate real life problem involving a multiphase transport-reaction equation We treat mobile and immobile pores in aporousmedia Such simulations aremade forwaste scenariossee [2]

We concentrate on the computational benefits of a fastcomputation of the commutators and their use for theinitialisation of the iterative splitting scheme

The equation is

1205971199051198881+ nabla sdot F119888

1= 119892 (minus119888

1+ 1198881119894119898

) minus 12058211198881 in Ω times [0 119905]

1205971199051198882+ nabla sdot F119888

2= 119892 (minus119888

2+ 1198882119894119898

) + 12058211198881minus 12058221198882 in Ω times [0 119905]

F = v minus 119863nabla

1205971199051198881119894119898

= 119892 (1198881minus 1198881119894119898

) minus 12058211198881119894119898

in Ω times [0 119905]

1205971199051198882119894119898

= 119892 (1198882minus 1198882119894119898

) + 12058211198881119894119898

minus 12058221198882119894119898

in Ω times [0 119905]

1198881(x 119905) = 119888

10(x) 119888

2(x 119905) = 119888

20(x) on Ω

1198881(x 119905) = 119888

11(x 119905) 119888

2(x 119905) = 119888

21(x 119905)

on 120597Ω times [0 119905]

1198881119894119898

(x 119905) = 0 1198882119894119898

(x 119905) = 0 on Ω

1198881119894119898 (x 119905) = 0 119888

2119894119898 (x 119905) = 0 on 120597Ω times [0 119905]

(41)

Here the parameters are V = 01 119863 = 001 1205821= 1205822= 01

and 119892 = 001We next treat the semidiscretised equation given by the

matrices

120597119905C = (

119860 minus Λ1minus 119866 0 119866 0

Λ1

119860 minus Λ2minus 119866 0 119866

119866 0 minusΛ1minus 119866 0

0 119866 Λ1

minusΛ2minus 119866

)C

(42)

where C = (c1 c2 c1119894119898 c2119894119898)119879 while c1 = (119888

11 119888

1119868) is the

solution of the first species in themobile phase in each spatialdiscretisation point (119894 = 1 119868) and the same for the othersolution vectors

We have the following two operators for the splittingmethod

119860 =119863

Δ1199092sdot (

minus2 1

1 minus2 1

d d d1 minus2 1

1 minus2

)

+VΔ119909

sdot (

1

minus1 1

d dminus1 1

minus1 1

) isin R119868times119868

(43)

where 119868 is the number of spatial pointsConsider

Λ1= (

1205821

0

0 1205821

0

d d d0 1205821

0

0 1205821

)

Λ2= (

1205822

0

0 1205822

0

d d d0 1205822

0

0 1205822

) isin R119868times119868

119866 = (

119892 0

0 119892 0

d d d0 119892 0

0 119892

) isin R119868times119868

(44)

We decouple this into the following matrices

119860 = (

119860 0 0 0

0 119860 0 0

0 0 0 0

0 0 0 0

) isin R4119868times4119868

1198611= (

minusΛ1

0 0 0

Λ1

minusΛ2

0 0

0 0 minusΛ1

0

0 0 Λ1

minusΛ2

)

1198612= (

minus119866 0 119866 0

0 minus119866 0 119866

119866 0 minus119866 0

0 119866 0 minus119866

) isin R4119868times4119868

(45)

For the operators 119860 and 119861 = 1198611+ 1198612and 119861 we apply the

iterative splitting method given in Section 21Based on the decomposition119860 is block diagonal and 119861 is

tridiagonalThe commutator is

[119860 119861] = (

0 0 119860119866 0

0 0 0 119860119866

minus119860119866 0 0 0

0 minus119860119866 0 0

) isin R4119868times4119868

(46)

10 International Journal of Differential Equations

Strang

AB

102

101

100

10minus1

10minus2

10minus3

100

Δt

c1

c2

c3

c4

c5

c6

Erro

r L1

AB Strang one-sided with B

(a)

CPU

tim

e

AB Strang one-side

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

(b)

Figure 5 Numerical errors and CPU time of the standard splittingscheme and the iterative schemes with Zassenhaus formula wherewe have at least 1 4 iterative steps

where we have a sparser operator than the main operator 119861and therefore we save computation time in computing suchoperators Further we assume that we have a bounded spatialstep size which is given by Δ119909 = 01 ≫ 0 so that we scale thesingular entry119863Δ119909

2asymp 1 and VΔ119909 asymp 1

Figure 5 presents the numerical error that is the differ-ence between the exact and the numerical solution and theCPU time required The optimal results are from the one-sided iterative schemes on the operator 119861

Remark 16 We obtain the same results as in the one-phase example because of the basically constant CPUtime for decreasing time steps of the iterative scheme weobtain optimal results when applying 1ndash3 Zassenhaus stepsTherefore with the embedded Zassenhaus formula we can

reach improved results more quickly than with the standardschemes With 2-3 more iterative steps we obtain moreaccurate results With the one-sided iterative schemes wereach the best convergence results when using the moredelicate operator 119861

4 Conclusions and Discussion

In this paper we have presented a novel splitting schemecombining the ideas of iterative and sequential schemes basedon the Zassenhaus formula Here the idea is to decouplethe expensive iterative computation of the schemes basedonly on the matrix exponential obtaining simpler embeddedZassenhaus schemes based on nilpotent commutators Suchcombinations have the benefit of reducing the computationtime because the commutators can be computed very cheaplyOn the other hand simple linear iterative steps can be donevery cheaply if they are initialised with a sufficiently accuratestarting solution and this accelerates the solvers The erroranalysis showed that these are stablemethods for higher orderschemes In the applications we were able to demonstratethe speedup from the Zassenhaus enhanced method andwere able to apply it to multiphysics applications In thefuture we will concentrate on linear and nonlinear matrix-dependent schemes that switch between noniterative anditerative schemes based on Zassenhaus products

References

[1] R E Ewing ldquoUp-scaling of biological processes andmultiphaseow in porous mediardquo in IIMA Volumes in Mathematics andIts Applications vol 295 pp 195ndash215 Springer New York NYUSA 2002

[2] J Geiser Discretization and Simulation of Systems forConvection-Diffusion-Dispersion Reactions with Applicationsin Groundwater Contamination Groundwater ModellingManagement and Contamination Nova Science MonographNew York NY USA 2008

[3] N Antonic C J van Duijn W Jager and A MikelicMultiscaleProblems in Science and Technology Challenges toMathematicalAnalysis and Perspectives Springer Berlin Germany 2002

[4] E Weinan Principle of Multiscale Modelling Cambridge Uni-versity Press Cambridge UK 2011

[5] S Blanes and F Casas ldquoOn the necessity of negative coefficientsfor operator splitting schemes of order higher than twordquoAppliedNumerical Mathematics vol 54 no 1 pp 23ndash37 2005

[6] EHansen andAOstermann ldquoHigh order splittingmethods foranalytic semigroups existrdquo BIT NumericalMathematics vol 49no 3 pp 527ndash542 2009

[7] G Strang ldquoOn the construction and comparison of differenceschemesrdquo SIAM Journal on Numerical Analysis vol 5 pp 506ndash517 1968

[8] G I Marchuk ldquoSome applicatons of splitting-up methods tothe solution of problems in mathematical physicsrdquo AplikaceMatematiky vol 1 pp 103ndash132 1968

[9] J Geiser ldquoComputing exponential for iterative splitting meth-ods algorithms and applicationsrdquo Journal of AppliedMathemat-ics vol 2011 Article ID 193781 27 pages 2011

[10] R I McLachlan and G RW Quispel ldquoSplittingmethodsrdquoActaNumerica vol 11 pp 341ndash434 2002

International Journal of Differential Equations 11

[11] J Geiser G Tanoglu and N Gucuyenen ldquoHigher order oper-ator splitting methods via Zassenhaus product formula theoryand applicationsrdquo Computers amp Mathematics with Applicationsvol 62 no 4 pp 1994ndash2015 2011

[12] G H Weiss and A A Maradudin ldquoThe Baker-Hausdorff for-mula and a problem in crystal physicsrdquo Journal of MathematicalPhysics vol 3 pp 771ndash777 1962

[13] R MWilcox ldquoExponential operators and parameter differenti-ation in quantum physicsrdquo Journal of Mathematical Physics vol8 pp 962ndash982 1967

[14] J Geiser ldquoAn iterative splitting approach for linear integro-differential equationsrdquo Applied Mathematics Letters vol 26 no11 pp 1048ndash1052 2013

[15] F Casas and A Murua ldquoAn efficient algorithm for computingthe Baker-Campbell-Hausdorff series and some of its applica-tionsrdquo Journal of Mathematical Physics vol 50 no 3 Article ID033513 2009

[16] J Geiser Iterative Splitting Methods for Differential EquationsChapmanampHallCRCNumerical Analysis and Scientific Com-puting CRC Press Boca Raton Fla USA 2011

[17] S Vandewalle Parallel Multigrid Waveform Relaxation forParabolic Problems Teubner Stuttgart Germany 1993

[18] J Geiser and G Tanoglu ldquoOperator-splitting methods viathe Zassenhaus product formulardquo Applied Mathematics andComputation vol 217 no 9 pp 4557ndash4575 2011

[19] F Bayen ldquoOn the convergence of the Zassenhaus formulardquoLetters in Mathematical Physics vol 3 no 3 pp 161ndash167 1979

[20] F Casas A Murua and M Nadinic ldquoEfficient computation ofthe Zassenhaus formulardquo Computer Physics Communicationsvol 183 no 11 pp 2386ndash2391 2012

[21] E Hairer C Lubich and GWannerGeometric Numerical Inte-gration vol 31 of Springer Series in Computational MathematicsSpringer Berlin Germany 2002

[22] H Yoshida ldquoConstruction of higher order symplectic integra-torsrdquo Physics Letters A vol 150 no 5ndash7 pp 262ndash268 1990

[23] E Faou A Ostermann and K Schratz ldquoAnalysis of exponentialsplitting methods for inhomogeneous parabolic equationsrdquohttparxivorgabs12125827

[24] J Geiser ldquoDiscretization methods with embedded analyticalsolutions for convection-diffusion dispersion-reaction equa-tions and applicationsrdquo Journal of EngineeringMathematics vol57 no 1 pp 79ndash98 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Differential Equations 9

calculating the nilpotent commutators in the Zassenhausformula So it makes sense to have 1ndash3 Zassenhaus stepsbefore starting the iterative scheme At that point only 2-3iterative steps are needed to obtainmore accurate results thanthose from the expensive standard schemes Here too weobtain the best results with the one-sided iterative schemes

Remark 15 Based on our restriction to 1198631 1198632 and Δ119909 with

(1198632minus 1198631)Δ1199092

asymp 1 we can control the delicate blowupof the computed commutators By the way if we neglectthe restriction and Δ119909 with (119863

2minus 1198631)Δ1199092

rarr infin thecommutator blows up and we lose the benefits of such analgorithmThis is a limitation of applying such an embeddedZassenhaus formula

322 Two Phase Example The next example is a moredelicate real life problem involving a multiphase transport-reaction equation We treat mobile and immobile pores in aporousmedia Such simulations aremade forwaste scenariossee [2]

We concentrate on the computational benefits of a fastcomputation of the commutators and their use for theinitialisation of the iterative splitting scheme

The equation is

1205971199051198881+ nabla sdot F119888

1= 119892 (minus119888

1+ 1198881119894119898

) minus 12058211198881 in Ω times [0 119905]

1205971199051198882+ nabla sdot F119888

2= 119892 (minus119888

2+ 1198882119894119898

) + 12058211198881minus 12058221198882 in Ω times [0 119905]

F = v minus 119863nabla

1205971199051198881119894119898

= 119892 (1198881minus 1198881119894119898

) minus 12058211198881119894119898

in Ω times [0 119905]

1205971199051198882119894119898

= 119892 (1198882minus 1198882119894119898

) + 12058211198881119894119898

minus 12058221198882119894119898

in Ω times [0 119905]

1198881(x 119905) = 119888

10(x) 119888

2(x 119905) = 119888

20(x) on Ω

1198881(x 119905) = 119888

11(x 119905) 119888

2(x 119905) = 119888

21(x 119905)

on 120597Ω times [0 119905]

1198881119894119898

(x 119905) = 0 1198882119894119898

(x 119905) = 0 on Ω

1198881119894119898 (x 119905) = 0 119888

2119894119898 (x 119905) = 0 on 120597Ω times [0 119905]

(41)

Here the parameters are V = 01 119863 = 001 1205821= 1205822= 01

and 119892 = 001We next treat the semidiscretised equation given by the

matrices

120597119905C = (

119860 minus Λ1minus 119866 0 119866 0

Λ1

119860 minus Λ2minus 119866 0 119866

119866 0 minusΛ1minus 119866 0

0 119866 Λ1

minusΛ2minus 119866

)C

(42)

where C = (c1 c2 c1119894119898 c2119894119898)119879 while c1 = (119888

11 119888

1119868) is the

solution of the first species in themobile phase in each spatialdiscretisation point (119894 = 1 119868) and the same for the othersolution vectors

We have the following two operators for the splittingmethod

119860 =119863

Δ1199092sdot (

minus2 1

1 minus2 1

d d d1 minus2 1

1 minus2

)

+VΔ119909

sdot (

1

minus1 1

d dminus1 1

minus1 1

) isin R119868times119868

(43)

where 119868 is the number of spatial pointsConsider

Λ1= (

1205821

0

0 1205821

0

d d d0 1205821

0

0 1205821

)

Λ2= (

1205822

0

0 1205822

0

d d d0 1205822

0

0 1205822

) isin R119868times119868

119866 = (

119892 0

0 119892 0

d d d0 119892 0

0 119892

) isin R119868times119868

(44)

We decouple this into the following matrices

119860 = (

119860 0 0 0

0 119860 0 0

0 0 0 0

0 0 0 0

) isin R4119868times4119868

1198611= (

minusΛ1

0 0 0

Λ1

minusΛ2

0 0

0 0 minusΛ1

0

0 0 Λ1

minusΛ2

)

1198612= (

minus119866 0 119866 0

0 minus119866 0 119866

119866 0 minus119866 0

0 119866 0 minus119866

) isin R4119868times4119868

(45)

For the operators 119860 and 119861 = 1198611+ 1198612and 119861 we apply the

iterative splitting method given in Section 21Based on the decomposition119860 is block diagonal and 119861 is

tridiagonalThe commutator is

[119860 119861] = (

0 0 119860119866 0

0 0 0 119860119866

minus119860119866 0 0 0

0 minus119860119866 0 0

) isin R4119868times4119868

(46)

10 International Journal of Differential Equations

Strang

AB

102

101

100

10minus1

10minus2

10minus3

100

Δt

c1

c2

c3

c4

c5

c6

Erro

r L1

AB Strang one-sided with B

(a)

CPU

tim

e

AB Strang one-side

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

(b)

Figure 5 Numerical errors and CPU time of the standard splittingscheme and the iterative schemes with Zassenhaus formula wherewe have at least 1 4 iterative steps

where we have a sparser operator than the main operator 119861and therefore we save computation time in computing suchoperators Further we assume that we have a bounded spatialstep size which is given by Δ119909 = 01 ≫ 0 so that we scale thesingular entry119863Δ119909

2asymp 1 and VΔ119909 asymp 1

Figure 5 presents the numerical error that is the differ-ence between the exact and the numerical solution and theCPU time required The optimal results are from the one-sided iterative schemes on the operator 119861

Remark 16 We obtain the same results as in the one-phase example because of the basically constant CPUtime for decreasing time steps of the iterative scheme weobtain optimal results when applying 1ndash3 Zassenhaus stepsTherefore with the embedded Zassenhaus formula we can

reach improved results more quickly than with the standardschemes With 2-3 more iterative steps we obtain moreaccurate results With the one-sided iterative schemes wereach the best convergence results when using the moredelicate operator 119861

4 Conclusions and Discussion

In this paper we have presented a novel splitting schemecombining the ideas of iterative and sequential schemes basedon the Zassenhaus formula Here the idea is to decouplethe expensive iterative computation of the schemes basedonly on the matrix exponential obtaining simpler embeddedZassenhaus schemes based on nilpotent commutators Suchcombinations have the benefit of reducing the computationtime because the commutators can be computed very cheaplyOn the other hand simple linear iterative steps can be donevery cheaply if they are initialised with a sufficiently accuratestarting solution and this accelerates the solvers The erroranalysis showed that these are stablemethods for higher orderschemes In the applications we were able to demonstratethe speedup from the Zassenhaus enhanced method andwere able to apply it to multiphysics applications In thefuture we will concentrate on linear and nonlinear matrix-dependent schemes that switch between noniterative anditerative schemes based on Zassenhaus products

References

[1] R E Ewing ldquoUp-scaling of biological processes andmultiphaseow in porous mediardquo in IIMA Volumes in Mathematics andIts Applications vol 295 pp 195ndash215 Springer New York NYUSA 2002

[2] J Geiser Discretization and Simulation of Systems forConvection-Diffusion-Dispersion Reactions with Applicationsin Groundwater Contamination Groundwater ModellingManagement and Contamination Nova Science MonographNew York NY USA 2008

[3] N Antonic C J van Duijn W Jager and A MikelicMultiscaleProblems in Science and Technology Challenges toMathematicalAnalysis and Perspectives Springer Berlin Germany 2002

[4] E Weinan Principle of Multiscale Modelling Cambridge Uni-versity Press Cambridge UK 2011

[5] S Blanes and F Casas ldquoOn the necessity of negative coefficientsfor operator splitting schemes of order higher than twordquoAppliedNumerical Mathematics vol 54 no 1 pp 23ndash37 2005

[6] EHansen andAOstermann ldquoHigh order splittingmethods foranalytic semigroups existrdquo BIT NumericalMathematics vol 49no 3 pp 527ndash542 2009

[7] G Strang ldquoOn the construction and comparison of differenceschemesrdquo SIAM Journal on Numerical Analysis vol 5 pp 506ndash517 1968

[8] G I Marchuk ldquoSome applicatons of splitting-up methods tothe solution of problems in mathematical physicsrdquo AplikaceMatematiky vol 1 pp 103ndash132 1968

[9] J Geiser ldquoComputing exponential for iterative splitting meth-ods algorithms and applicationsrdquo Journal of AppliedMathemat-ics vol 2011 Article ID 193781 27 pages 2011

[10] R I McLachlan and G RW Quispel ldquoSplittingmethodsrdquoActaNumerica vol 11 pp 341ndash434 2002

International Journal of Differential Equations 11

[11] J Geiser G Tanoglu and N Gucuyenen ldquoHigher order oper-ator splitting methods via Zassenhaus product formula theoryand applicationsrdquo Computers amp Mathematics with Applicationsvol 62 no 4 pp 1994ndash2015 2011

[12] G H Weiss and A A Maradudin ldquoThe Baker-Hausdorff for-mula and a problem in crystal physicsrdquo Journal of MathematicalPhysics vol 3 pp 771ndash777 1962

[13] R MWilcox ldquoExponential operators and parameter differenti-ation in quantum physicsrdquo Journal of Mathematical Physics vol8 pp 962ndash982 1967

[14] J Geiser ldquoAn iterative splitting approach for linear integro-differential equationsrdquo Applied Mathematics Letters vol 26 no11 pp 1048ndash1052 2013

[15] F Casas and A Murua ldquoAn efficient algorithm for computingthe Baker-Campbell-Hausdorff series and some of its applica-tionsrdquo Journal of Mathematical Physics vol 50 no 3 Article ID033513 2009

[16] J Geiser Iterative Splitting Methods for Differential EquationsChapmanampHallCRCNumerical Analysis and Scientific Com-puting CRC Press Boca Raton Fla USA 2011

[17] S Vandewalle Parallel Multigrid Waveform Relaxation forParabolic Problems Teubner Stuttgart Germany 1993

[18] J Geiser and G Tanoglu ldquoOperator-splitting methods viathe Zassenhaus product formulardquo Applied Mathematics andComputation vol 217 no 9 pp 4557ndash4575 2011

[19] F Bayen ldquoOn the convergence of the Zassenhaus formulardquoLetters in Mathematical Physics vol 3 no 3 pp 161ndash167 1979

[20] F Casas A Murua and M Nadinic ldquoEfficient computation ofthe Zassenhaus formulardquo Computer Physics Communicationsvol 183 no 11 pp 2386ndash2391 2012

[21] E Hairer C Lubich and GWannerGeometric Numerical Inte-gration vol 31 of Springer Series in Computational MathematicsSpringer Berlin Germany 2002

[22] H Yoshida ldquoConstruction of higher order symplectic integra-torsrdquo Physics Letters A vol 150 no 5ndash7 pp 262ndash268 1990

[23] E Faou A Ostermann and K Schratz ldquoAnalysis of exponentialsplitting methods for inhomogeneous parabolic equationsrdquohttparxivorgabs12125827

[24] J Geiser ldquoDiscretization methods with embedded analyticalsolutions for convection-diffusion dispersion-reaction equa-tions and applicationsrdquo Journal of EngineeringMathematics vol57 no 1 pp 79ndash98 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 International Journal of Differential Equations

Strang

AB

102

101

100

10minus1

10minus2

10minus3

100

Δt

c1

c2

c3

c4

c5

c6

Erro

r L1

AB Strang one-sided with B

(a)

CPU

tim

e

AB Strang one-side

AB

Δt

Strang

100

10minus1

10minus2

10minus3

10minus4

10minus2

10minus1

100

c1

c2

c3

c4

c5

c6

(b)

Figure 5 Numerical errors and CPU time of the standard splittingscheme and the iterative schemes with Zassenhaus formula wherewe have at least 1 4 iterative steps

where we have a sparser operator than the main operator 119861and therefore we save computation time in computing suchoperators Further we assume that we have a bounded spatialstep size which is given by Δ119909 = 01 ≫ 0 so that we scale thesingular entry119863Δ119909

2asymp 1 and VΔ119909 asymp 1

Figure 5 presents the numerical error that is the differ-ence between the exact and the numerical solution and theCPU time required The optimal results are from the one-sided iterative schemes on the operator 119861

Remark 16 We obtain the same results as in the one-phase example because of the basically constant CPUtime for decreasing time steps of the iterative scheme weobtain optimal results when applying 1ndash3 Zassenhaus stepsTherefore with the embedded Zassenhaus formula we can

reach improved results more quickly than with the standardschemes With 2-3 more iterative steps we obtain moreaccurate results With the one-sided iterative schemes wereach the best convergence results when using the moredelicate operator 119861

4 Conclusions and Discussion

In this paper we have presented a novel splitting schemecombining the ideas of iterative and sequential schemes basedon the Zassenhaus formula Here the idea is to decouplethe expensive iterative computation of the schemes basedonly on the matrix exponential obtaining simpler embeddedZassenhaus schemes based on nilpotent commutators Suchcombinations have the benefit of reducing the computationtime because the commutators can be computed very cheaplyOn the other hand simple linear iterative steps can be donevery cheaply if they are initialised with a sufficiently accuratestarting solution and this accelerates the solvers The erroranalysis showed that these are stablemethods for higher orderschemes In the applications we were able to demonstratethe speedup from the Zassenhaus enhanced method andwere able to apply it to multiphysics applications In thefuture we will concentrate on linear and nonlinear matrix-dependent schemes that switch between noniterative anditerative schemes based on Zassenhaus products

References

[1] R E Ewing ldquoUp-scaling of biological processes andmultiphaseow in porous mediardquo in IIMA Volumes in Mathematics andIts Applications vol 295 pp 195ndash215 Springer New York NYUSA 2002

[2] J Geiser Discretization and Simulation of Systems forConvection-Diffusion-Dispersion Reactions with Applicationsin Groundwater Contamination Groundwater ModellingManagement and Contamination Nova Science MonographNew York NY USA 2008

[3] N Antonic C J van Duijn W Jager and A MikelicMultiscaleProblems in Science and Technology Challenges toMathematicalAnalysis and Perspectives Springer Berlin Germany 2002

[4] E Weinan Principle of Multiscale Modelling Cambridge Uni-versity Press Cambridge UK 2011

[5] S Blanes and F Casas ldquoOn the necessity of negative coefficientsfor operator splitting schemes of order higher than twordquoAppliedNumerical Mathematics vol 54 no 1 pp 23ndash37 2005

[6] EHansen andAOstermann ldquoHigh order splittingmethods foranalytic semigroups existrdquo BIT NumericalMathematics vol 49no 3 pp 527ndash542 2009

[7] G Strang ldquoOn the construction and comparison of differenceschemesrdquo SIAM Journal on Numerical Analysis vol 5 pp 506ndash517 1968

[8] G I Marchuk ldquoSome applicatons of splitting-up methods tothe solution of problems in mathematical physicsrdquo AplikaceMatematiky vol 1 pp 103ndash132 1968

[9] J Geiser ldquoComputing exponential for iterative splitting meth-ods algorithms and applicationsrdquo Journal of AppliedMathemat-ics vol 2011 Article ID 193781 27 pages 2011

[10] R I McLachlan and G RW Quispel ldquoSplittingmethodsrdquoActaNumerica vol 11 pp 341ndash434 2002

International Journal of Differential Equations 11

[11] J Geiser G Tanoglu and N Gucuyenen ldquoHigher order oper-ator splitting methods via Zassenhaus product formula theoryand applicationsrdquo Computers amp Mathematics with Applicationsvol 62 no 4 pp 1994ndash2015 2011

[12] G H Weiss and A A Maradudin ldquoThe Baker-Hausdorff for-mula and a problem in crystal physicsrdquo Journal of MathematicalPhysics vol 3 pp 771ndash777 1962

[13] R MWilcox ldquoExponential operators and parameter differenti-ation in quantum physicsrdquo Journal of Mathematical Physics vol8 pp 962ndash982 1967

[14] J Geiser ldquoAn iterative splitting approach for linear integro-differential equationsrdquo Applied Mathematics Letters vol 26 no11 pp 1048ndash1052 2013

[15] F Casas and A Murua ldquoAn efficient algorithm for computingthe Baker-Campbell-Hausdorff series and some of its applica-tionsrdquo Journal of Mathematical Physics vol 50 no 3 Article ID033513 2009

[16] J Geiser Iterative Splitting Methods for Differential EquationsChapmanampHallCRCNumerical Analysis and Scientific Com-puting CRC Press Boca Raton Fla USA 2011

[17] S Vandewalle Parallel Multigrid Waveform Relaxation forParabolic Problems Teubner Stuttgart Germany 1993

[18] J Geiser and G Tanoglu ldquoOperator-splitting methods viathe Zassenhaus product formulardquo Applied Mathematics andComputation vol 217 no 9 pp 4557ndash4575 2011

[19] F Bayen ldquoOn the convergence of the Zassenhaus formulardquoLetters in Mathematical Physics vol 3 no 3 pp 161ndash167 1979

[20] F Casas A Murua and M Nadinic ldquoEfficient computation ofthe Zassenhaus formulardquo Computer Physics Communicationsvol 183 no 11 pp 2386ndash2391 2012

[21] E Hairer C Lubich and GWannerGeometric Numerical Inte-gration vol 31 of Springer Series in Computational MathematicsSpringer Berlin Germany 2002

[22] H Yoshida ldquoConstruction of higher order symplectic integra-torsrdquo Physics Letters A vol 150 no 5ndash7 pp 262ndash268 1990

[23] E Faou A Ostermann and K Schratz ldquoAnalysis of exponentialsplitting methods for inhomogeneous parabolic equationsrdquohttparxivorgabs12125827

[24] J Geiser ldquoDiscretization methods with embedded analyticalsolutions for convection-diffusion dispersion-reaction equa-tions and applicationsrdquo Journal of EngineeringMathematics vol57 no 1 pp 79ndash98 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Differential Equations 11

[11] J Geiser G Tanoglu and N Gucuyenen ldquoHigher order oper-ator splitting methods via Zassenhaus product formula theoryand applicationsrdquo Computers amp Mathematics with Applicationsvol 62 no 4 pp 1994ndash2015 2011

[12] G H Weiss and A A Maradudin ldquoThe Baker-Hausdorff for-mula and a problem in crystal physicsrdquo Journal of MathematicalPhysics vol 3 pp 771ndash777 1962

[13] R MWilcox ldquoExponential operators and parameter differenti-ation in quantum physicsrdquo Journal of Mathematical Physics vol8 pp 962ndash982 1967

[14] J Geiser ldquoAn iterative splitting approach for linear integro-differential equationsrdquo Applied Mathematics Letters vol 26 no11 pp 1048ndash1052 2013

[15] F Casas and A Murua ldquoAn efficient algorithm for computingthe Baker-Campbell-Hausdorff series and some of its applica-tionsrdquo Journal of Mathematical Physics vol 50 no 3 Article ID033513 2009

[16] J Geiser Iterative Splitting Methods for Differential EquationsChapmanampHallCRCNumerical Analysis and Scientific Com-puting CRC Press Boca Raton Fla USA 2011

[17] S Vandewalle Parallel Multigrid Waveform Relaxation forParabolic Problems Teubner Stuttgart Germany 1993

[18] J Geiser and G Tanoglu ldquoOperator-splitting methods viathe Zassenhaus product formulardquo Applied Mathematics andComputation vol 217 no 9 pp 4557ndash4575 2011

[19] F Bayen ldquoOn the convergence of the Zassenhaus formulardquoLetters in Mathematical Physics vol 3 no 3 pp 161ndash167 1979

[20] F Casas A Murua and M Nadinic ldquoEfficient computation ofthe Zassenhaus formulardquo Computer Physics Communicationsvol 183 no 11 pp 2386ndash2391 2012

[21] E Hairer C Lubich and GWannerGeometric Numerical Inte-gration vol 31 of Springer Series in Computational MathematicsSpringer Berlin Germany 2002

[22] H Yoshida ldquoConstruction of higher order symplectic integra-torsrdquo Physics Letters A vol 150 no 5ndash7 pp 262ndash268 1990

[23] E Faou A Ostermann and K Schratz ldquoAnalysis of exponentialsplitting methods for inhomogeneous parabolic equationsrdquohttparxivorgabs12125827

[24] J Geiser ldquoDiscretization methods with embedded analyticalsolutions for convection-diffusion dispersion-reaction equa-tions and applicationsrdquo Journal of EngineeringMathematics vol57 no 1 pp 79ndash98 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of