NUMERICAL SMOOTHNESS AND ERROR ANALYSIS ON WENO FOR THE NONLINEAR

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NUMERICAL SMOOTHNESS AND ERROR ANALYSIS ONWENO FOR THE NONLINEAR CONSERVATION LAWS

TONG SUN

Abstract. In this study we give an a posteriori error analysis on the WENO

schemes for the nonlinear scalar conservation laws. This analysis is based onthe new concept of numerical smoothness, with some new error analysis mecha-

nisms developed for the finite difference and finite volume discretizations. The

local error estimate is of optimal order in space and time. The global errorestimate grows linearly in time, because of the direct application of the L1-

contraction between entropy solutions in the error propagation analysis. As a

beginning, we only deal with smooth solutions in this paper. Within the sameerror propagation framework, when we deal with piecewise smooth solutions

later, we only need to work on estimating the local error where smoothness is

lost. The smoothness indicators not only serve the purpose of local error esti-mation, but also serve as a monitor on both the possible numerical instability

and the expected solution shapening.

1. Introduction

The Weighted Essentially NonOscillatory schemes (WENO) are widely acceptedas a successful method for solving nonlinear conservation laws, because it presentshigh order accuracy for smooth solutions and essentially no oscillation at shocks.Due to the complexity of the nonlinear equations and schemes, there has been verylimited error analysis results for WENO. In [2], Jiang and Shu proved that a WENOscheme is convergent for smooth solutions. This proof is based on Strang’s theory[7] on the stability of the first variation of a scheme. The stability of the firstvariation of a WENO scheme depends on the smoothness of the solution, hence itis not easy to generalize the convergence result to discontinuous solutions. To ourbest knowledge, there has not been any a posteriori error analysis result on WENOschemes. Some general a posteriori error analysis results for conservation laws canbe found in [1] and [3].

In this paper, we present a new a posteriori error analysis for WENO. The ideabehind this new error analysis is to use the smoothness of a numerical solution toestimate its local error, and to use the L1-contraction between entropy solutions toestimate error propagation. We have applied the idea successfully to the RKDGschemes in [8]. This is our first attempt to apply the idea to a finite difference orfinite volume method, hence some new error analysis mechanisms are developed.

Received by the editors October 3, 2011.

2010 Mathematics Subject Classification. 65M15 .

c©0000 (copyright holder)

1

2 TONG SUN

In general, an error analysis of an initial value problem consists of a local er-ror estimation and an error propagation estimation. Most a priori error analy-sis rely on the smoothness of exact PDE solutions to estimate local error, conse-quently one has to estimate how schemes propagate error. The error propagationof a scheme is usually given by a von Neumann stability condition of the form‖u1(t + ∆t) − u2(t + ∆t)‖ ≤ (1 + α∆t)‖u1(t) − u2(t)‖, where u1 and u2 are twonumerical solutions satisfying the same scheme. For a nonlinear conservation lawsolved by a WENO scheme, it is extremely hard, if not impossible, to establish avon Neumann condition with α near 0. However, in general, a large α would leadto an exponentially growing coefficient eαt in the error estimate, which could makethe estimate meaningless for affordable ∆x and ∆t. In contrary to the error prop-agation of a complex scheme, while solving a scalar conservation law, it is trivial touse the L1-contraction between entropy solutions for error propagation. However, adirect consequence of considering error propagation between PDE solutions is thatone has to consider local error around numerical solutions. As a matter of fact,local error can only be considered along numerical solutions in a posteriori erroranalysis. Perhaps because most numerical solutions are technically not smooth,residual error has been employed in most cases. When schemes are relatively sim-ple, such as a streamline diffusion scheme in [3], residual error works well. In orderto deal with more sophisticated schemes and get sharper local error estimates, webelieve that it is necessary to work with the smoothness of numerical solutions.

Although numerical solutions of PDEs are usually not smooth functions, goodnumerical solutions do resemble the smoothness and discontinuities of the corre-sponding PDE solutions. For example, smoothness indicators are computed inthe ENO and WENO schemes. In parallel with the phrase “numerical stability”,it is natural to refer what is indicated by smoothness indicators as “numericalsmoothness”, which means the smoothness of numerical solutions. Such numericalsmoothness indicators have been used in adaptive scheme designs in many cases,such as recently in [4]. In our recent work, numerical smoothness with new indi-cators are investigated for the purpose of error estimation, which seems to be in arelatively new research area in the literature of numerical PDEs.

As expected, it is significantly more complex to analyze the local error by usingthe smoothness of a numerical solution. Nevertheless, there are several advantagesof using numerical smoothness, including: (1) The smoothness of a numerical so-lution is quantitatively known through the computation of smoothness indicators,hence the error estimate will be sharp. (2) When we work with the computednumerical solutions, we have direct access to L∞-norm estimates on smoothnessand error, which is crucial to the nonlinear problems. (3) The error propagationbetween PDE solutions is actually how error should be propagated, so the obtainederror estimate is optimal in terms of error propagation. (4) In time-dependentPDEs, nonlinearity causes difficulties. To be more precise, it is very hard to dealwith global properties, but much easier to deal with local properties. With thehelp of smoothness indicators, which inform us if smoothness has been maintainedover time before tn, we only need to study smoothness locally within a time step[tn, tn+1] and use it for local error estimation. Regardless of the tediousness, thetask is typically not as hard as impossible. (5) Numerical smoothness indicatorsalso serve as “instability” detectors. If/when a smoothness indicator unexpectedlybecomes large or oscillating at a point in the space-time domain, the scheme is

NUMERICAL SMOOTHNESS AND ERROR ANALYSIS ON WENO 3

probably turning unstable or anti-diffusive over there. An early diagnosis shouldgive us a chance to adaptively treat the problem before the entire solution is ruined.

It is worthwhile to elaborate on point (4) of the last paragraph. When we doerror analysis for a time dependent PDE, we have to deal with global propertieshere or there. Error propagation and smoothness maintenance are both globalproperties, we have to deal with one of them in terms of the PDE, the other onein terms of the numerical scheme. In other words, we have two alternative choices.First, we can rely on the PDE’s global smoothness maintenance to do local errorestimation and deal with the global error propagation of our numerical scheme;or second, we can rely on the PDE’s global error propagation and deal with thescheme’s global smoothness maintenance for local error estimation. The secondchoice is not popular yet, but it is better on both error propagation estimation andlocal error estimation:

• Error Propagation. If we have an error propagation rate estimate for anyconvergent numerical scheme on a PDE problem, we can pass the estimateto limit and get an equally good error propagation rate estimate for the PDEitself. Hence, an error propagation rate estimate is available for the PDEwhenever an error propagation rate estimate is available for any one of itsnumerical schemes. Moreover, the PDE’s error propagation rate is equal toor better than the error propagation rate of any related numerical scheme.In the case of scalar conservation laws, we go with the L1-contraction, whichcan be obtained by using the Lax-Friedrich scheme.

• Local Error. Although we only use those schemes which are designed tocarry a right level of numerical diffusion and total variation control, theglobal smoothness maintenance behavior of a scheme is certainly hard toprove. However, we can get access to the numerical smoothness needed forlocal error estimation in two steps. The first step is the computation ofthe smoothness indicators at a time tn. Although we usually cannot provethe boundedness of these smoothness indicators, we can always computethem to see if they are reasonably bounded. The second step is to quan-titatively establish the smoothness of certain auxiliary PDE solutions andsemi-discrete solutions within a time step [tn, tn+1], based on the computedsmoothness indicators at tn. This is usually doable because it is a local re-sult. If we were using the smoothness of an exact PDE solution, we wouldnot have sharp quantitative smoothness estimates in space and time. Thetwo step numerical smoothness establishment enables us to do sharp localerror estimation.

The computation of smoothness indicators is how we bypass the typically over-whelming difficulty caused by nonlinearity. If we were dealing with a scheme’s errorpropagation, we would have no easy way to bypass the difficulty. The fundamentaldifference is that numerical smoothness is merely a property of the computed nu-merical solution, while numerical error propagation involves not only the computedsolution, but also another non-computable solution satisfying the same numericalscheme.

As the beginning step of tackling WENO, we only deal with “smooth” numericalsolutions in this paper. Namely, we work within the scenario where the computedsmoothness indicators are bounded by a constant M . Our error propagation es-timation remains valid for discontinuous solutions, because L1-contraction is valid

4 TONG SUN

between any pair of entropy solutions. When we deal with non-smooth solutions inthe future, we only need to work on the local error analysis where smoothness hasbeen lost. The current analysis and results are applicable to the smooth pieces of anon-smooth solution. The local error analysis is to estimate how well an auxiliaryentropy solution is approximated in a step, which is a safe approach. Moreover,the detailed cell by cell and step by step local error analysis for the smooth casehas also given us abundant information about the self-sharpening process and itseffect on the local error growth. Therefore we now have a good staring point in theinvestigation of the local error where smoothness has deteriorated.

In the next section, we present the semi-discrete WENO scheme and the sub-sequent TVD-Runge-Kutta time stepping scheme. In section 3, we first define areconstructed numerical solution and a few auxiliary solutions of the PDE and thesemi-discrete WENO scheme. These auxiliary solutions make it possible to splitand estimate the error of the WENO-TVDRK scheme. In the rest of section 3, wedefine the smoothness indicators to be computed out of a numerical solution. Theseindicators will enter the error estimates, playing the role of high order derivatives.In section 4, we prove the main error estimation theorem through a sequence oflemmas. In section 5, we present some numerical evidences for the boundedness ofthe smoothness indicators, which validate the error analysis. In the last section, wemake some conclusion remarks.

2. WENO schemes

For the convenience of the reader, we give the following description of the WENOscheme [5]. In this paper, we only present the results for the WENO scheme of order5 (in space). Consider the one-dimensional nonlinear conservation law

(2.1) ut + f(u)x = 0

in a bounded interval Ω = [a, b]. In order to focus on the new ideas and the newtools of the proof, we stay with the simple case of west wind ( f ′(u) > 0 ) and theperiodic boundary condition

u(t, a) = u(t, b)for all t ≥ 0. Let the initial condition be

(2.2) u(0, x) = uI(x).

Assume that the flux function f(u) is sufficiently smooth and the initial functionuI(x) is a smooth periodic function, which guarantees that the entropy solutionu(t, x) is smooth for (t, x) ∈ [0, T ]× (−∞,∞), where T is a positive time.

Partition Ω with a = x1/2 < x3/2 < · · · < xM+1/2 = b. Let h = xi+ 12− xi− 1

2be

the same for all cells Ωi = (xi− 12, xi+ 1

2). Because we are considering the problems

under the periodic boundary condition, as spatial cell and node indices, we identifyM + 1/2 + δ with 1/2 + δ, and 1/2 − δ with M + 1/2 − δ, for any δ ≥ 0. Wealso partition the time domain by 0 = t0 < t1 < · · · < tN = T . For simplicity, weassume that τ = tn+1 − tn is constant for n = 0, 1, · · · , N − 1.

Without loss of generality, we assume that all the solutions involved in the discus-sion are bounded in L∞-norm. Under this assumption, we can define the followingtwo constants.

β = max f ′(u), δ = max |f ′′(u)|.Consequently, we require that the CFL condition is satisfied, βτ ≤ h.


In order to define a numerical solution obtained by using the WENO scheme,we use un,i to denote the average of the numerical solution in the cell Ωi at timetn. Collecting all the cell averages, we have un = (un,1, · · · , un,M ) as the numericalsolution at time tn. At the initial time t = 0, the initial values of the numericalsolution is given by the cell averages of the initial function uI(x),

u0,i =1h

∫Ωi

uI(x) dx.

In the WENO scheme, a semi-discrete (method of line) solution satisfies

(2.3)d

dtuh

n,i = − 1h

[fi+ 1

2(uh

n)− fi− 12(uh

n)],

where the numerical flux f is to be determined by the WENO reconstruction (be-low). The fully discrete solution un+1 is computed by solving (2.3) with a TVD-Runge-Kutta scheme, such as (2.13) .

To the end of formulating the semi-discrete WENO scheme (2.3), we start witha typical cell Ωi, and the three stencils

Si,2 = (Ωi−2,Ωi−1,Ωi), Si,1 = (Ωi−1,Ωi,Ωi+1), Si,0 = (Ωi,Ωi+1,Ωi+2).

Consider the reconstruction of the cell boundary values of a function v(x) from itscell averages (v1, · · · , vM ), given by the three different stencils. In each stencil Si,r

(r = 2, 1, 0), define a quadratic polynomial pi,r by

1h

∫Ωi−r+q

pi,r(x) dx = vi−r+q, q = 0, 1, 2.

The candidates of order 3 ENO reconstructions are v(r)

i+ 12

= pi,r(xi+ 12) and v

(r)

i− 12

=pi,r(xi− 1

2), r = 0, 1, 2. It is shown in [5] that

v(2)

i+ 12

=13vi−2 −

76vi−1 +

116

vi, v(2)

i− 12

= −16vi−2 +

56vi−1 +

13vi,

v(1)

i+ 12

= −16vi−1 +

56vi +

13vi+1, v

(1)

i− 12

=13vi−1 +

56vi −

16vi+1,

v(0)

i+ 12

=13vi +

56vi+1 −

16vi+2, v

(0)

i− 12

=116

vi −76vi+1 +

13vi+2.

A WENO reconstruction is given by a weighted summation of the candidateENO reconstructions from these stencils. The key to the success of a WENOscheme would be the choice of the weights ωi,r for each cell Ωi. For the sake ofstability and consistency, it is required that

(2.4) ωi,r ≥ 0,2∑

r=0

ωi,r = 1.

If a function v(x) is smooth in all of the candidate stencils, by choosing the constantsdr with

d0 =310

, d1 =610

, d2 =110

,

we have

(2.5) vi+ 12

=2∑

r=0

drv(r)

i+ 12

= v(xi+ 12) + O(h5).

6 TONG SUN

If the function to be reconstructed is smooth in all the stencils, it is ideal to haveωi,r = dr + O(h2).

When the function v(x) has a discontinuity in one or more of the stencils Si,r, itis desirable that the corresponding weights ωi,r to be essentially 0, to emulate thesuccessful ENO idea. Based on these and some other considerations, in [2], Jiangand Shu proposed the following form of weights:

(2.6) ωi,r = ωi,r(vi−2, vi−1, vi, vi+1, vi+2) =αr∑2

s=0 αs

with

(2.7) αr =dr

(ε + βr)2.

Here ε > 0 is introduced to avoid the denominator to become 0. We take ε = h4 inall our numerical tests. βr is the “smoothness indicator” of the stencil Si,r:

β0 =1312

(vi − 2vi+1 + vi+2)2 +14(3vi − 4vi+1 + vi+2)2,

β1 =1312

(vi−1 − 2vi + vi+1)2 +14(vi−1 − vi+1)2,

β2 =1312

(vi−2 − 2vi−1 + vi)2 +14(vi−2 − 4vi−1 + 3vi)2.(2.8)

By symmetry, for r = 0, 1, 2, let dr = d2−r, αr = dr

(ε+βr)2 , ωi,r = αr∑2s=0 αs

. Finally,the fifth order WENO reconstructions at the end points of Ωi are

(2.9) v−i+ 1

2=

2∑r=0

ωi,rv(r)

i+ 12

=2∑

r=−2

ηi,rvi+r,

(2.10) v+i− 1

2=

2∑r=0

ωi,rv(r)

i− 12

=2∑

r=−2

ηi,rvi+r,

where ηi,−2 = 13ωi,2, ηi,−1 = − 7

6ωi,2 − 16ωi,1, ηi,0 = 11

6 ωi,2 + 56ωi,1 + 1

3ωi,0,ηi,1 = 1

3ωi,1 + 56ωi,0, ηi,2 = − 1

6ωi,0, and the formulas for ηi,r can be derived inthe same way. In the error analysis, we also need the notation ηd

r , which means thevalue of ηi,r with ωi,q replaced by dq, for r = −2,−1, 0, 1, 2, q = 0, 1, 2. Obviously,

ηd−2 =

260

, ηd−1 = −13

60, ηd

0 =4760

, ηd1 =

2760

, ηd2 = − 3

60.

For simplicity, under the “west wind” assumption f ′(u) > 0, we use the Godunovnumerical flux

fi+ 12(v) = f(v−

i+ 12).

This is the formula for the numerical flux in the the semi-discrete problem (2.3).In this special case, d, α, ω and v+

i− 12

are actually not employed.For temporal discretization, we take a standard TVD-RK scheme of order k [5].

For example, when k = 3, in the time step [tn, tn+1], we compute the fully-discretesolution un+1 from un by the following scheme. With the notation

Hi(v) = − 1h

[fi+ 1

2(v)− fi− 1

2(v)],


the scheme is that

(2.11) uS1n,i = un,i + τHi(un),

(2.12) uS2n,i =

34un,i +

14uS1

n,i +τ

4Hi(uS1

n ),

(2.13) un+1,i =13un,i +

23uS2

n,i +2τ

3Hi(uS2

n ).

In the computation of the function Hi(·) in (2.11), (2.12) , (2.13) and later in thedefinition of the semi-discrete solution uH

n (t), within each time step [tn, tn+1], theweights ωi,r are computed from un, and they remain constant in [tn, tn+1].

3. Error splitting and numerical smoothness

In order to estimate the global error of a WENO-TVDRK solution by using itsnumerical smoothness, we need to introduce a few auxiliary solutions. With theseauxiliary solutions, we can split the global error into propagated error, spatialdiscretization error and temporal discretization error.

3.1. Auxiliary solutions. First, we define a standard 6-cell reconstruction. Sinceerror propagation will be estimated in the L1-norm, we need to reconstruct a solu-tion function from the cell averages of the numerical solution. In general, if we knowthe cell averages (v1, · · · , vM ) of a function v(x), we define a piecewise polynomialfunction vR(x) to be the reconstruction. For a typical cell Ωi = (xi− 1

2, xi+ 1

2), we

first define the local reconstruction vR,i(x) by using the cell averages in the six cellsnear xi− 1

2. vR,i(x) is the polynomial of degree 5 satisfying

1h

∫Ωi+r

vR,i(x) dx = vi+r, r = −3,−2,−1, 0, 1, 2.

If we expend vR,i(x) at xi− 12

by

vR,i(x) = ai(x− xi− 12)5 + bi(x− xi− 1

2)4 + ci(x− xi− 1

2)3(3.1)

+ di(x− xi− 12)2 + ei(x− xi− 1

2) + fi,

it is not too hard to verify that the coefficients can be computed by

ai =1

120h5(−vi−3 + 5vi−2 − 10vi−1 + 10vi − 5vi+1 + vi+2),

bi =1

48h4(vi−3 − 3vi−2 + 2vi−1 + 2vi − 3vi+1 + vi+2),

ci =1

36h3(vi−3 − 11vi−2 + 28vi−1 − 28vi + 11vi+1 − vi+2),

di =1

16h2(−vi−3 + 7vi−2 − 6vi−1 − 6vi + 7vi+1 − vi+2),

ei =1

180h(−2vi−3 + 25vi−2 − 245vi−1 + 245vi − 25vi+1 + 2vi+2),

fi =160

(vi−3 − 8vi−2 + 37vi−1 + 37vi − 8vi+1 + vi+2).

8 TONG SUN

We can also rewrite vR,i(x) as

vR,i(x) = vi−3R−3(x− xi− 12) + vi−2R−2(x− xi− 1

2) + vi−1R−1(x− xi− 1

2)

+ viR0(x− xi− 12) + vi+1R1(x− xi− 1

2) + vi+2R2(x− xi− 1

2),

where, from (3.1),

R−3(x) = − x5

120h5+

x4

48h4+

x3

36h3− x2

16h2− x

90h+

160

,

R−2(x) =x5

24h5− x4

16h4− 11x3

36h3+

7x2

16h2+

5x

36h− 2

15,

R−1(x) = − x5

12h5+

x4

24h4+

7x3

9h3− 3x2

8h2− 49x

36h+

3760

,

R0(x) =x5

12h5+

x4

24h4− 7x3

9h3− 3x2

8h2+

49x

36h+

3760

,

R1(x) = − x5

24h5− x4

16h4+

11x3

36h3+

7x2

16h2− 5x

36h− 2

15,

R2(x) =x5

120h5+

x4

48h4− x3

36h3− x2

16h2+

x

90h+

160

.

After defining vR,i(x) for each cell index i, we are ready to define the wholedomain reconstruction vR(x), which consists of piecewise polynomials of degree 5,

vR(x) = vR,i(x), x ∈ Ωi.

To reuse this reconstruction procedure to different cases, we define the linear re-construction operators R and Ri,

vR(x) = R(v1, · · · , vM ), vR,i(x) = Ri(vi−3, vi−2, vi−1, vi, vi+1, vi+2).

With the help of the reconstruction operators R and Ri, we are ready to define theneeded auxiliary solutions:

• Reconstructed numerical solution uRn (x) at each time tn

The first application of the reconstruction operator R is to convert the cell aver-ages of a numerical solution at each time tn to a discontinuous piecewise polynomialfunction,

uRn (x) = R(un,1, · · · , un,M ).

With this reconstruction, we consider the global error of the numerical solution as

‖u(tn, x)− uRn (x)‖L1(Ω), n = 1, · · · , N.

The spatial smoothness indicator will be defined below in terms of uRn (x).

• Entropy solution uEn (t, x) in each time step [tn, tn+1]

In each time step [tn, tn+1], we consider an entropy solution of the conservationlaw (2.1), denoted by uE

n (t, x), satisfying the initial condition uEn (tn, x) = uR

n (x).Moreover, we consider the cell averages of uE

n (t, x),

uEn,i(t) =

1h

∫Ωi

uEn (t, x)dx, i = 1, · · · ,M

and the subsequent reconstruction

uERn (t, x) = R(uE

n,1(t), · · · , uEn,M (t)).


• Semi-discrete WENO solution uHn,i(t) in [tn, tn+1], i = 1, · · · ,M

In each time step [tn, tn+1], we consider a semi-discrete WENO solution uHn (t) =

(uHn,1(t), · · · , uH

n,M (t)) satisfying (2.3). That is,

(3.2)d

dtuH

n,i = − 1h

[fi+ 1

2(uH

n )− fi− 12(uH

n )], i = 1, · · · ,M.

The Godunov flux under the assumption f ′(u) > 0 implies that

(3.3) fi+ 12(uH

n ) = f(u−n,i+ 1

2),

where u−n,i+ 1

2is the linear combination of uH

n,i−2, uHn,i−1, u

Hn,i, u

Hn,i+1 and uH

n,i+2 givenby (2.9), in which each ωi,r is computed by (2.6). The initial condition at tn is

uHn,i(tn) = un,i, i = 1, · · · ,M.

From the cell averages uHn,i(t), we can define the reconstruction

uHRn (t, x) = R(uH

n,1(t), · · · , uHn,M (t)).

• Overlapping local strong solutions uL,in (t, x) in [tn, tn+1], i = 1, · · · ,M

Since the entropy solution uEn (t, x) is almost never smooth in the interior of

any cell, we need to introduce a local strong solution uL,in (t, x) of (2.1) in Ui =

∪2r=−3Ωi+r. The initial condition at time tn is, for x ∈ Ui ∪ Ωi−4,

uL,in (tn, x) = uR,i

n (x) = Ri(un,i−3, un,i−2, un,i−1, un,i, un,i+1, un,i+2).

When the standard CFL condition is satisfied, by the characteristic line theory,uL,i

n (t, x) is well defined in [tn, tn+1] × Ui, provided τ ≤ 1/(δ max |uR,ix |). We also

defineuL,i

n,i+r(t) =1h

∫Ωi+r

uL,in (t, x) dx r = −3,−2,−1, 0, 1, 2,

and

uL,i,Rn (t, x) = Ri(u

L,in,i−3(t), u

L,in,i−2(t), u

L,in,i−1(t), u

L,in,i(t), u

L,in,i+1(t), u

L,in,i+2(t)).

3.2. Error splitting. In this paper, we estimate the L1-norm error between theexact solution u(tn, x) and reconstructed numerical solution uR

n (x) at each timetn. Based on the auxiliary solutions defined above, we can split the global erroru(tn+1, x)− uR

n+1(x) as in the following graph:

uRn

u(tn)uR

n+1

uHRn (tn+1)

uEn (tn+1)

u(tn+1)

-

6

tn tn+1

In the graph and also sometimes in the rest of the paper, we hide one of the twoindependent variables in the notation of a solution to make the expressions shorter.

10 TONG SUN

The error analysis of this paper is based on the error splitting in the graph. InL1-norm, we have

‖u(tn+1)− uRn+1‖L1(Ω) ≤ ‖u(tn+1)− uE

n (tn+1)‖L1(Ω)(3.4)

+ ‖uEn (tn+1)− uHR

n (tn+1)‖L1(Ω)

+ ‖uHRn (tn+1)− uR

n+1‖L1(Ω).

The first part u(tn+1)−uEn (tn+1) is the propagation of the global error u(tn)−uR

n

by the PDE. Due to the L1-contraction property of the scalar conservation laws [6],since both u and uE

n are |Ω|-periodic, it is easy to prove

(3.5) ‖u(tn+1)− uEn (tn+1)‖L1(Ω) ≤ ‖u(tn)− uR

n ‖L1(Ω).

The second part of the split error uEn (tn+1) − uHR

n (tn+1) is the local spatialdiscretization error. To estimate this part of the error, we need the smoothness ofuE

n (t, x). Since its initial value uRn (x) lives in a discontinuous piecewise polynomial

space, uEn (t, x) is certainly not smooth in the classical sense. However, we have

found out how to quantitatively define and establish its numerical smoothness, andhow to use the obtained smoothness on local error estimation.

The third part of the split error uHRn (tn+1) − uR

n+1 is the local temporal dis-cretization error. To estimate this part of the error, we will need the temporalsmoothness of uH

n (t) for t ∈ [tn, tn+1]. We have found out how to establish itssmoothness, with the stiffness taken into account. The subsequent local error esti-mation for the Runga-Kutta solution is standard.

Here it is to be noticed that, since both uEn (t, x) and uH

n (t) depend on the nu-merical solution uR

n (x) or un as their initial value, the smoothness level of themdepends on how un has been computed. From the numerical results, it is clear thatWENO-TVDRK does maintain the numerical smoothness. The Godunov upwind-ing flux, the WENO design and the TVD property of the Runge-Kutta schememust have provided sufficient smoothness maintenance. If we were able to proveit, we would have a priori error estimates. The most important advantage of thenumerical smoothness approach of error analysis is that we can always computethe smoothness indicators to obtain the needed smoothness information, and use itback to a posteriori error estimation.

3.3. Smoothness indicators. In order to rigorously and quantitatively define theconcept of numerical smoothness for the WENO-TVDRK scheme, we define thefollowing spatial and temporal smoothness indicators at each time tn, which is tobe used for the local error analysis in the time step [tn, tn+1]. In the definitions,S stands for space, T stands for time, k is the order of the Runge-Kutta scheme.Since we only consider the 5-cell stencil WENO scheme, the sup-index 5 means thatthe reconstruction polynomials uR,i

n (x) are of degree 5.

• Definition of the spatial smoothness indicator S5n

The spatial smoothness indicator S5n contains not only the spatial derivatives of

uRn (x) within each cell, but also the jumps of the derivatives of uR

n (x) across thecell boundaries. Namely, for the cell Ωi = (xi− 1

2, xi+ 1

2),

S5n,i = (M0

n,i,M1n,i,M

2n,i,M

3n,i,M

4n,i,M

5n,i, D

0n,i, D

1n,i, D

2n,i, D

4n,i, D

4n,i, D

5n,i),

where

M ln,i =

dl

dxluR

n (x+i− 1

2), Ll

n,i =dl

dxluR

n (x−i− 1

2),


J ln,i = M l

n,i − Lln,i, Dl

n,i =J l

n,i

h6−l.

Based on the reconstruction formula (3.1), it is easy to explicitly compute all thecomponents of S5

n,i. Obviously, the values of M ln,i and Ll

n,i should be of O(1),unless there is a shock or contact discontinuity somewhere around Ωi. It is alsoeasy to guess that the jumps J l

n,i should be small, otherwise the numerical solutionmay have lost too much smoothness at the cell boundary. How small should thejumps J l

n,i be? Both our error analysis and numerical experiments suggest thatDl

n,i = J ln,i/h6−l should be at most of O(1), unless there is a shock or high order

discontinuity within or near the cell. This is the reason for having Dln,i instead of

J ln,i serving as a part of the smoothness indicator. The boundedness of Dl

n,i, l =0, · · · , 5, reflects the smoothness of uR

n (x) at the cell boundary xi−1/2, which is atechnical discontinuity due to piecewise approximation.

• Definition of the temporal smoothness indicator T kn

The temporal smoothness indicator T kn consists of the temporal derivatives of

uHn at t = tn. Namely,

T kn =

(un,

d

dtuH

n (tn),d2

dt2uH

n (tn), · · · ,dk+1

dtk+1uH

n (tn))

.

The first derivative ddtu

Hn (tn) is computed as in the implementation of the forward

Euler scheme

(3.6)d

dtuH

n,i(tn) = Hi(un).

The formula for computing the second derivative can be obtained by taking deriva-tives with respect to time on both sides of the semi-discrete scheme (3.2) and (3.3):

(3.7)d2

dt2uH

n,i(tn) =1h

[f ′(u−

n,i− 12)

d

dtu−

n,i− 12− f ′(u−

n,i+ 12)

d

dtu−

n,i+ 12

].

Since u−n,i+ 1

2is a linear combination of uH

n,i−2, uHn,i−1, u

Hn,i, u

Hn,i+1 and uH

n,i+2 given by

(2.9), d2

dt2 uHn (tn) can be computed by (3.7) and the previously computed d

dtuHn (tn)

in (3.6).In the same way of computing d2

dt2 uHn,i(tn), one can compute d3

dt3 uHn,i(tn) and

d4

dt4 uHn,i(tn). If the popular 3-rd order TVD-RK scheme (2.13) is to be used, it is

enough to have the fourth derivative. When the flux is nonlinear, the chain rulewill make the formulas for higher order derivatives more tedious. However, onecan successively compute these derivatives, using the result of d

dtuHn (tn) in (3.6)

and d2

dt2 uHn (tn) in (3.7) for the computation of d3

dt3 uHn (tn), and so on. The cost of

computing all these temporal derivatives are proportional to that of implementingthe Runge-Kutta scheme.

Remark. The amount of work for computing S5n is proper. In fact, S5

n containsthe minimal amount of smoothness information for us to estimate the optimalapproximation error of the piecewise polynomials of degree 5 to the PDE solution.The amount of work for computing T k

n is also proper, for the same reason. It mightbe possible to estimate T k

n from S5n (if k and 5 are related in certain way), but a

directly computed T kn should sharpen the estimate of the temporal local error.

12 TONG SUN

4. Error analysis

In this section, we present the error analysis results. We try to give sufficientdetails in the proofs to show the boundedness and computability of all the parametersin the error estimates, but avoid tediousness and redundancy. More details are tobe presented in a technical report. The first lemma is useful in a few places.

Lemma 4.1. If (v1, · · · , vM ) consists of the cell averages of a function v(x) ineach Ωi (i = 1, · · · ,M), and vR(x) = R(v1, · · · , vM ), then there is a computableconstant C1, such that

‖vR(x)‖L1(Ω) ≤ C1hM∑i=1

|vi|.

Proof. In Ωi, by the definition of the reconstruction,

vR(x) = vi−3R−3(x− xi− 12) + vi−2R−2(x− xi− 1

2) + vi−1R−1(x− xi− 1

2)

+ viR0(x− xi− 12) + vi+1R1(x− xi− 1

2) + vi+2R2(x− xi− 1

2).

Let

I−3 =1h

∫ h

0

|R−3(x)| dx ≈ 0.00976563, I−2 =1h

∫ h

0

|R−2(x)| dx ≈ 0.0723323,

I−1 =1h

∫ h

0

|R−1(x)| dx ≈ 0.272618, I0 =1h

∫ h

0

|R0(x)| dx = 1,

I1 =1h

∫ h

0

|R1(x)| dx ≈ 0.129824, I2 =1h

∫ h

0

|R2(x)| dx ≈ 0.0140252.

Then, it is obvious that

‖vR(x)‖L1(Ωi) ≤ h (I−3|vi−3|+ I−2|vi−2|+ I−1|vi−1|+ I0|vi|+ I1|vi+1|+ I2|vi+2|).

Consequently, by the periodicity,

‖vR(x)‖L1(Ω) ≤M∑i=1

‖vR(x)‖L1(Ωi) ≤ h (I−3 + I−2 + I−1 + I0 + I1 + I2)M∑i=1

|vi|.

Here one can choose C1 = 1.5 > I−3 + I−2 + I−1 + I0 + I1 + I2. #

The numerical smoothness given by the spatial smoothness indicator S5n can be

used to estimate the local spatial discretization error. The proof of this mostlydepends on the local strong solutions uL,i

n (t, x). We have the following lemmas topresent the smoothness and approximation properties of uL,i

n (t, x).

Lemma 4.2. There is a computable function C2, such that, for i = 1, · · · ,M ,

(4.1) ‖uEn (tn+1, x)− uL,i

n (tn+1, x)‖L1(Ωi) ≤ τh6C2(S5n,i).

Proof. At time tn, uEn (tn, x) = uR

n (x) = uL,in (tn, x) for x ∈ Ωi. However, in Ωi−1,

uEn (tn, x) and uL,i

n (tn, x) are not equal. In fact, in Ωi−1,

uL,in (tn, x) = uR,i

n (x) = M0n,i + M1

n,i(x− xi− 12) +

M2n,i

2(x− xi− 1

2)2

+M3

n,i

6(x− xi− 1

2)3 +

M4n,i

24(x− xi− 1

2)4 +

M5n,i

120(x− xi− 1

2)5,


but

uEn (tn, x) = uR,i−1

n (x) = L0n,i + L1

n,i(x− xi− 12) +

L2n,i

2(x− xi− 1

2)2

+L3

n,i

6(x− xi− 1

2)3 +

L4n,i

24(x− xi− 1

2)4 +

L5n,i

120(x− xi− 1

2)5.

Under the CFL condition βτ ≤ h, for any x ∈ (xi− 12− βτ, xi− 1

2) ⊂ Ωi−1,

|uL,in (tn, x)− uE

n (tn, x)| =

∣∣∣∣∣5∑

l=0

J ln,i

l!(x− xi− 1

2)l

∣∣∣∣∣ .Now, due to the L1-contraction property between different entropy solutions (see[6], Chapter 16),

‖uEn (tn+1, x)− uL,i

n (tn+1, x)‖L1(Ωi)

≤ ‖uEn (tn, x)− uL,i

n (tn, x)‖L1(xi− 12−βτ,x

i+ 12)

= ‖uEn (tn, x)− uL,i

n (tn, x)‖L1(xi− 12−βτ,x

i− 12)

≤ (βτ) h65∑

l=0

|Dln,i|

(l + 1)!(βτ/h)l.

The Lemma is proven with C2(S5n,i) = β

∑5l=0

|Dln,i|

(l+1)! (βτ/h)l.#

Remark: We have kept the quotient βτ/h in the constants. In numerical experi-ments, we found that the quotient needs to be fairly small. Keeping the quotient inthe constants makes the constants smaller, which is a gain for the error estimates.

Lemma 4.3. There are computable functions Nk, for k = 0, 1, 2, 3, 4, 5, such that

(4.2)∣∣∣∣ ∂k

∂xkuL,i

n (t, x)∣∣∣∣ ≤ Nk(S5

n,i)

for all x ∈ Ui and t ∈ [tn, tn+1]. Moreover, there is a computable function N6, suchthat

(4.3)∣∣∣∣ ∂6

∂x6uL,i

n (t, x)∣∣∣∣ ≤ τN6(S5

n,i).

Proof. For k = 0, since the strong solution maintains its extreme values, and

uL,in (tn, x) =

5∑l=0

M ln,i

l!(x− xi− 1

2)l,

we get ∣∣uL,in (t, x)

∣∣ ≤ N0(S5n) =

5∑l=0

|M ln,i|l!

(4h)l.

To estimate the upper bounds of the derivatives, we start by giving them shortnotations used only in this proof. Let

wk =∂k

∂xkuL,i

n (t, x), k = 0, 1, 2, 3, 4, 5, 6.

14 TONG SUN

As a strong solution of the conservation law, w0(t, x) = uL,in (t, x) satisfies

∂

∂tw0 + f ′(w0)

∂

∂xw0 = 0.

Taking derivative with respect to x from the last equation, we get

(4.4)∂

∂tw1 + f ′(w0)

∂

∂xw1 + f ′′(w0)w2

1 = 0.

By considering ∂∂t + f ′(w0) ∂

∂x as the derivative along the characteristics, we canestimate the point values of w1(t, x). Since initially at time tn

w1(tn, x) =d

dxuR,i

n (x) = M1n,i + · · ·+

M5n,i

(4)!(x− xi− 1

2)4,

w1(tn, x) is a bounded function of S5n,i. By solving (4.4) along a characteristic

x = x∗ + f ′(w0(tn, x∗))(t− tn), we get

w1(t, x) =w1(tn, x∗)

1 + f ′′(w0(tn, x∗))w1(tn, x∗)(t− tn).

When δτ |w1(tn, x∗)| ≤ 1/2 (not a restrictive condition for a smooth solution),

(4.5) |w1(t, x)| ≤ |w1(tn, x∗)|(1 + 2δτ |w1(tn, x∗)|) ≤ N1(S5n,i),

if N1(S5n,i) = maxx∗∈Ωi−4∪Ui

|w1(tn, x∗)|(1+2δτ |w1(tn, x∗)|). Next we consider thefollowing linear equations obtained from differentiating (4.4),

∂

∂tw2 + f ′(w0)

∂

∂xw2 + 3f ′′(w0)w1w2 + f ′′′(w0)w3

1 = 0,

∂

∂tw3 + f ′(w0)

∂

∂xw3 + 4f ′′(w0)w1w3 + F3(w0, w1, w2) = 0,

∂

∂tw4 + f ′(w0)

∂

∂xw4 + 5f ′′(w0)w1w4 + F4(w0, w1, w2, w3) = 0,

∂

∂tw5 + f ′(w0)

∂

∂xw5 + 6f ′′(w0)w1w5 + F5(w0, w1, w2, w3, w4) = 0,

∂

∂tw6 + f ′(w0)

∂

∂xw6 + 7f ′′(w0)w1w6 + F6(w0, w1, w2, w3, w4, w5) = 0,(4.6)

where the function F3, F4, F5 and F6 can be easily written out, although theyare somewhat tedious. By solving these linear differential equations along thecharacteristics, we can recursively bound w2, w3, w4 and w5 by some computablefunctions of M0

n,i,M1n,i,M

2n,i,M

3n,i,M

4n,i and M5

n,i. Thus we have proven (4.2). Itis important to notice that

(4.7) w6(tn, x) = 0.

From (4.6) and (4.7), we can derive and solve a differential inequality, which willlead us to prove (4.3). #

Lemma 4.4. There is a computable function CB,i,r for each pair of integers r ∈−4,−3,−2,−1, 0, 1, 2 and i ∈ 1, · · · ,M, such that

|uL,in (t, xi+r+1/2)− uE

n (t, xi+r+1/2)| ≤ h6CB,i,r(S5n), t ∈ (tn, tn+1].

The function CB,i,r(S5n) actually only depends on S5

n,r for r = −3,−2,−1, 0, 1, 2.


Proof. We start by looking at the difference |uL,in (tn, x) − uE

n (tn, x)| cell by cell.In Ωi, we know that uL,i

n (tn, x) = uEn (tn, x). In Ωi−1, |uL,i

n (tn, x) − uEn (tn, x)| =∑5

k=0 |Jk

n,i

k! (x−xi− 12)k| ≤ h6CB,−1(S5

n,i), for a computable function CB,−1. In Ωi−2,

|uL,in (tn, x)− uE

n (tn, x)| = |uR,in (x)− uR,i−2

n (x)|≤ |uR,i

n (x)− uR,i−1n (x)|+ |uR,i−1

n (x)− uR,i−2n (x)|

=5∑

k=0

|Jk

n,i

k!(x− xi− 1

2)k|+

5∑k=0

|Jk

n,i−1

k!(x− xi−3/2)k|

≤ h6CB,−2(S5n,i−1, S

5n,i)

for a computable function CB,−2. Similarly, we can bound |uL,in (tn, x)−uE

n (tn, x)| inΩi−3 by h6CB,−3(S5

n,i−2, S5n,i−1, S

5n,i), in Ωi−4 by h6CB,−4(S5

n,i−3, S5n,i−2, S

5n,i−1, S

5n,i),

in Ωi+1 by h6CB,1(S5n,i+1), and in Ωi+2 by h6CB,2(S5

n,i+1, S5n,i+2).

Now, we can to estimate |uL,in (t, xi+r+1/2) − uE

n (t, xi+r+1/2)| for t ∈ (tn, tn+1].Let

A = uL,in (t, xi+r+1/2), B = uE

n (t, xi+r+1/2).

By using the characteristics, we have

A = uL,in (tn, xi+r+1/2 − f ′(A)(t− tn))

andB = uE

n (tn, xi+r+1/2 − f ′(B)(t− tn)).

Thus,

|A−B| = |uL,in (tn, xi+r+ 1

2− f ′(A)(t− tn))− uE

n (tn, xi+r+ 12− f ′(B)(t− tn))|

≤ |uL,in (tn, xi+r+ 1

2− f ′(A)(t− tn))− uL,i

n (tn, xi+r+ 12− f ′(B)(t− tn))|

+ |uL,in (tn, xi+r+ 1

2− f ′(B)(t− tn))− uE

n (tn, xi+r+ 12− f ′(B)(t− tn))|

≤ N1(S5n,i)δ(t− tn)|A−B|+ h6CB,i,r,

where CB,i,r is given by the function value of CB,r defined above. Thus

(4.8) |A−B| ≤ 11−N1(S5

n,i)δ(t− tn)CB,i,rh

6 ≤ [1 + 2N1(S5n,i)δτ ]h6CB,i,r,

if δτN1(S5n,i) ≤ 1/2. The lemma is proven with

CB,i,r(S5n) = [1 + 2N1(S5

n,i)δτ ] CB,i,r. #

Remark. For a nonlinear conservation law (δ > 0), the self-sharpening (N1(S5n,i) ≥

|ux| → ∞) worsens the smoothness and the local error. The inequalities (4.5) and(4.8) take this into account locally and sharply.

Lemma 4.5. There is a computable function CA,i,r for each pair of integers r ∈−3,−2,−1, 0, 1, 2 and i ∈ 1, · · · ,M, such that

|uL,in,i+r(t)− uE

n,i+r(t)| ≤ (t− tn)h5CA,i,r(S5n), t ∈ [tn, tn+1].

Proof. At time tn, from the definition of uL,in and uE

n , we know

uL,in,i+r(tn) = uE

n,i+r(tn) = un,i+r

16 TONG SUN

for r = −3,−2,−1, 0, 1, 2, hence

(4.9) uL,in,i+r(tn)− uE

n,i+r(tn) = 0.

As a strong solution, uL,in (t, x) satisfies

d

dtuL,i

n,i+r(t) =1h

[ f(uL,in (t, xi+r−1/2))− f(uL,i

n (t, xi+r+1/2))].

Because of the CFL condition and f ′(u) > 0, for t ∈ (tn, tn+1], as an entropysolution, uE

n (t, x) satisfiesd

dtuE

n,i+r(t) =1h

[ f(uEn (t, xi+r−1/2))− f(uE

n (t, xi+r+1/2))].

Taking the difference of the last two equations,d

dt[uL,i

n,i+r(t)− uEn,i+r(t)] =

1h

[ f(uL,in (t, xi+r−1/2))− f(uE

n (t, xi+r−1/2))]

− 1h

[ f(uL,in (t, xi+r+1/2))− f(uE

n (t, xi+r+1/2))].

By Lemma 4.4,∣∣∣∣ d

dt[uL,i

n,i+r(t)− uEn,i+r(t)]

∣∣∣∣ ≤ h5β[CB,i,r−1(S5

n) + CB,i,r(S5n)].

Taking (4.9) into account, we get∣∣∣uL,in,i+r(t)− uE

n,i+r(t)∣∣∣ ≤ (t− tn)h5β

[CB,i,r−1(S5

n) + CB,i,r(S5n)].

The lemma is proven with

CA,i,r(S5n) = βCB,i,r−1(S5

n) + βCB,i,r(S5n). #

Next we estimate the error between uEn (tn+1, x) and uER

n (tn+1, x).

Lemma 4.6. There is a computable function CE, such that

‖uEn (tn+1, x)− uER

n (tn+1, x)‖L1(Ω) ≤ τh5CE(S5n).

Proof. First, decompose the L1-norm into individual cells:


n (tn+1, x)‖L1(Ω) =M∑i=1


n (tn+1, x)‖L1(Ωi).

Now, with the help of the local strong solution uL,in , we can split the L1-norm error

in each cell Ωi by

‖uEn (t, x)− uER

n (t, x)‖L1(Ωi) ≤ ‖uEn (t, x)− uL,i

n (t, x)‖L1(Ωi)(4.10)

+ ‖uL,in (t, x)− uL,i,R

n (t, x)‖L1(Ωi)

+ ‖uL,i,Rn (t, x)− uER

n (t, x)‖L1(Ωi).

From Lemma 4.2,

(4.11) ‖uEn (t, x)− uL,i

n (t, x)‖L1(Ωi) ≤ τh6C2(S5n,i).

By the elementary interpolation theory, and Lemma 4.3, there is a computableconstant CR, such that

|uL,in (t, x)− uL,i,R

n (t, x)| ≤ CRh6 max∣∣∣∣ ∂6

∂x6uL,i

n (t, x)∣∣∣∣ ≤ τh6CRN6(S5

n,i).


Consequently,

(4.12) ‖uL,in (t, x)− uL,i,R

n (t, x)‖L1(Ωi) ≤ τh7CRN6(S5n,i).

It remains to estimate the last part of (4.10). As in the proof of Lemma 4.1,

‖uL,i,Rn (t, x)− uER

n (t, x)‖L1(Ωi) =

∥∥∥∥∥2∑

r=−3

(uL,in,i+r(t)− uE

n,i+r(t))Rr(x)

∥∥∥∥∥L1(Ωi)

≤2∑

r=−3

hIr|uL,in,i+r(t)− uE

n,i+r(t)|.

By Lemma 4.5, we get

(4.13) ‖uL,i,Rn (t, x)− uER

n (t, x)‖L1(Ωi) ≤ (t− tn)h62∑

r=−3

IrCA,i,r(S5n).

Combining (4.11), (4.12) and (4.13), we get


n (tn+1, x)‖L1(Ωi)(4.14)

≤ τh6

[C2(S5

n,i) + CRhN6(S5n,i) +

2∑r=−3

IrCA,i,r(S5n)

].

Finally,


n (tn+1, x)‖L1(Ω)(4.15)

≤ τh5M∑i=1

h

[C2(S5


2∑r=−3

IrCA,i,r(S5n)

].

The lemma is proven with

CE(S5n) =

M∑i=1

h

[C2(S5


2∑r=−3

IrCA,i,r(S5n)

]. #

In order to estimate the spatial local error for the semi-discrete WENO scheme,we give the following lemma.

Lemma 4.7. Let v(x) be a function in C[−3, 3], and vj be the average of v(x) in[j, j + 1], for j = −3,−2,−1, 0, 1 and 2. If v(x) ∈ C3[−3, 3], then, for some realnumbers ξr ∈ (−r, 3− r), r = 2, 1, 0,∣∣∣∣v(1)−

(13v−2 −

76v−1 +

116

v0

)∣∣∣∣ ≤ |v′′′(ξ2)|,∣∣∣∣v(1)−(−1

6v−1 +

56v0 +

13v1

)∣∣∣∣ ≤ 13|v′′′(ξ1)|,∣∣∣∣v(1)−

(13v0 +

56v1 −

16v2

)∣∣∣∣ ≤ 13|v′′′(ξ0)|.

If v(x) ∈ C4[−3, 3], then, for some real numbers ξr ∈ (−1− r, 3− r), r = 2, 1, 0,∣∣∣∣v(1)− v(0)−(

13v−2 −

76v−1 +

116

v0

)+(

13v−3 −

76v−2 +

116

v−1

)∣∣∣∣ ≤ 54|v(4)(ξ2)|,

18 TONG SUN∣∣∣∣v(1)− v(0)−(−1

6v−1 +

56v0 +

13v1

)+(−1

6v−2 +

56v−1 +

13v0

)∣∣∣∣ ≤ 512|v(4)(ξ1)|,∣∣∣∣v(1)− v(0)−

(13v0 +

56v1 −

16v2

)+(

13v−1 +

56v0 −

16v1

)∣∣∣∣ ≤ 512|v(4)(ξ0)|.

If v(x) ∈ C5[−3, 3], then, for a real number ξ ∈ (−2, 3),∣∣∣∣∣v(1)−2∑

r=−2

ηdrvr

∣∣∣∣∣ ≤ 110|v(5)(ξ)|.

If v(x) ∈ C6[−3, 3], then for a real number ξ ∈ (−3, 3),∣∣∣∣∣v(1)− v(0)−2∑

r=−2

ηdrvr +

2∑r=−2

ηdrvr−1

∣∣∣∣∣ ≤ 760|v(6)(ξ)|.

Proof. The proof is elementary. #

Lemma 4.8. There is a computable function CH , such that

‖uERn (tn+1, x)− uHR

n (tn+1, x)‖L1(Ω) ≤ τh5CH(S5n).

Proof. We start by looking into the cell averages of uERn (t, x) and uHR

n (t, x). Since

d

dtuE

n,i(t) =1h

[ f(uEn (t, xi− 1

2))− f(uE

n (t, xi+ 12))]

andd

dtuH

n,i(t) =1h

[fi− 1

2(uH

n (t))− fi+ 12(uH

n (t))],

we have ∣∣∣∣ d

dt

[uE

n,i(t)− uHn,i(t)

]∣∣∣∣ ≤ 1h

∣∣∣ f(uEn (t, xi− 1

2))− f(uL,i

n (t, xi− 12))∣∣∣(4.16)

+1h

∣∣∣ f(uEn (t, xi+ 1

2))− f(uL,i

n (t, xi+ 12))∣∣∣

+1h

[f(uL,i

n (t, xi+ 12))− fi+ 1

2(uL,i

n (t))]

−[f(uL,i

n (t, xi− 12))− fi− 1

2(uL,i

n (t))]

+1h

∣∣∣ fi+ 12(uL,i

n (t))− fi+ 12(uE

n (t))∣∣∣

+1h

∣∣∣ fi− 12(uL,i

n (t))− fi− 12(uE

n (t))∣∣∣

+1h

∣∣∣ fi+ 12(uE

n (t))− fi+ 12(uH

n (t))∣∣∣

+1h

∣∣∣ fi− 12(uE

n (t))− fi− 12(uH

n (t))∣∣∣ .

To analyze the RHS in this differential inequality, we start from the easier terms:

• By Lemma 4.4,∣∣∣ f(uE

n (t, xi− 12))− f(uL,i

n (t, xi− 12))∣∣∣ ≤ h6βCB,i,−1(S5

n).

• In the term in the second row,∣∣∣ f(uE

n (t, xi+ 12))− f(uL,i

n (t, xi+ 12))∣∣∣ = 0.


• Considering the way the Godunov flux is defined, by Lemma 4.5, we canfind a computable function C3, such that the two differences in the 5th and6th rows are bounded by h6C3(S5

n,i−3, S5n,i−2, S

5n,i−1, S

5n,i, S

5n,i+1, S

5n,i+2).

We omit the detail here because it is elementary and tedious.• Just by considering the way the Godunov flux is defined, we can bound

the two differences in the 7th and 8th rows by a linear combination of∣∣uEn,i+r(t)− uH

n,i+r(t)∣∣, r = −3,−2,−1, 0, 1 and 2. It is easy to compute the

coefficients of the linear combination, which should show that the coeffi-cients only depend on the WENO reconstruction weights.

The only two rows left are the third and fourth row, where we are looking at thelocal spatial discretization error of the right hand side of a semi-discrete WENOscheme on a smooth solution, uL,i

n (t, x). It is easy to show that each bracket of thesecond row is of O(h5) with the well known fact that ωi,r = dr +O(h2), r = 0, 1, 2.However, we need to show that the entire absolute value is of the form h6CD(S5

n,i),where the function CD needs to be computable. To this end, we will need to estitateωi,r−dr = O(h2) and ωi,r−ωi−1,r = O(h3). Since we are working with a posteriorierror analysis, instead of estimating these quantities in terms of S5

n, we wouldrather directly compute µi,r = ωi,r−dr

h2 , νi,r = ωi,r−ωi−1,r

h3 , µi = (µi,0, µi,1, µi,2), andνi = (νi,0, νi,1, νi,2), then using them in the error estimates. Now,

∣∣∣ [f(uL,in (t, xi+ 1

2))− fi+ 1

2(uL,i

n (t))]−[f(uL,i

n (t, xi− 12))− fi− 1

2(uL,i

n (t))]∣∣∣

=

[f(uL,i

n (t, xi+ 12))− f

(2∑

r=−2

ηi,ruL,in,i+r(t)

)]

−

[f(uL,i

n (t, xi− 12))− f

(2∑

r=−2

ηi−1,ruL,in,i−1+r(t)

)]

= f ′(θi)

[uL,i

n (t, xi+ 12)−

2∑r=−2

ηi,ruL,in,i+r(t)

]

−f ′(θi−1)

[uL,i

n (t, xi− 12)−

2∑r=−2


]

≤ | f ′(θi−1)| uL,in (t, xi+ 1

2)−

2∑r=−2

ηi,ruL,in,i+r(t)

−uL,in (t, xi− 1

2) +

2∑r=−2


+ | f ′(θi)− f ′(θi−1)|

∣∣∣∣∣uL,in (t, xi+ 1

2)−

2∑r=−2

ηi,ruL,in,i+r(t)

∣∣∣∣∣ .Here θi is in between uL,i

n (t, xi+ 12) and

∑2r=−2 ηi,ru

L,in,i+r(t). It is obvious that

|f ′(θi−1)| ≤ β. It is not hard to show that |f ′(θi) − f ′(θi−1)| ≤ hδCZ(S5n,i) for a

computable function CZ .

20 TONG SUN

To make the formulas brief, we introduce the following notations for the fre-quently used differences. For v = (v1, · · · , vM ), we define

Ii,2(v) =13vi−2 −

76vi−1 +

116

vi, Ii,1(v) = −16vi−1 +

56vi +

13vi+1,

and

Ii,0(v) =13vi +

56vi+1 −

16vi+2.

Now, by using Lemma 4.7 and the standard scaling argument, we have∣∣∣∣∣uL,in (t, xi+ 1

2)−

2∑r=−2

ηi,r uL,in,i+r(t)

∣∣∣∣∣≤

∣∣∣∣∣uL,in (t, xi+ 1

2)−

2∑r=−2

ηdr uL,i

n,i+r(t)

∣∣∣∣∣+∣∣∣∣∣

2∑r=−2

(ηr,i − ηdr )uL,i

n,i+r(t)

∣∣∣∣∣≤ h5

10N5(S5

n,i) +∣∣∣(ωi,2 − d2)

[uL,i

n (t, xi+ 12)− Ii,2(uL,i

n (t))]∣∣∣

+∣∣∣(ωi,1 − d1)

[uL,i

n (t, xi+ 12)− Ii,1(uL,i

n (t))]∣∣∣

+∣∣∣(ωi,0 − d0)

[uL,i

n (t, xi+ 12)− Ii,0(uL,i

n (t))]∣∣∣

≤ h5

10N5(S5

n,i) + h5µi,2N3(S5n,i) +

h5

3µi,1N3(S5

n,i) +h5

3µi,0N3(S5

n,i).

Similarly, without showing all the details, we have∣∣∣∣∣uL,in (t, xi+ 1

2)−

2∑r=−2

ηi,ruL,in,i+r(t)− uL,i

n (t, xi− 12) +

2∑r=−2


∣∣∣∣∣≤

∣∣∣∣∣uL,in (t, xi+ 1

2)−

2∑r=−2

ηdr uL,i

n,i+r(t)− uL,in (t, xi− 1

2) +

2∑r=−2

ηdr uL,i

n,i−1+r(t)

∣∣∣∣∣+

∣∣∣∣∣2∑

r=−2

(ηi,r − ηdr )uL,i

n,i+r(t)−2∑

r=−2

(ηi−1,r − ηdr )uL,i

n,i−1+r(t)

∣∣∣∣∣≤ 7

60τh6N6(S5

n,i) + (ωi,2 − d2)[uL,i

n (t, xi+ 12)− Ii,2(uL,i

n (t))]

−(ωi−1,2 − d2)[uL,i

n (t, xi− 12)− Ii−1,2(uL,i

n (t))]

+ (ωi,1 − d1)[uL,i

n (t, xi+ 12)− Ii,1(uL,i

n (t))]

−(ωi−1,1 − d1)[uL,i

n (t, xi− 12)− Ii−1,1(uL,i

n (t))]

+ (ωi,0 − d0)[uL,i

n (t, xi+ 12)− Ii,0(uL,i

n (t))]

−(ωi−1,0 − d0)[uL,i

n (t, xi− 12)− Ii−1,0(uL,i

n (t))]

≤ 760

τh6N6(S5n,i) + h6N4(S5

n,i)(

54µi,2 +

512

µi,1 +512

µi,0

)+h6N3(S5

n,i)(

νi,2 +13νi,1 +

13νi,0

).


Combining the estimates after (4.16), we know that there is a computable functionCEH , such that, for i = 1, · · · ,M ,∣∣∣∣ d

dt

[uE

n,i(t)− uHn,i(t)

]∣∣∣∣ ≤ h5CEH(S5n, µi, νi) +

β

h

2∑r=−2

|ηi,r|∣∣uE

n,i+r(t)− uHn,i+r(t)

∣∣+

β

h

2∑r=−2

|ηi−1,r|∣∣uE

n,i−1+r(t)− uHn,i−1+r(t)

∣∣ .By solving this system of differential inequalities in [tn, tn+1], noticing that uE

n,i(tn)−uH

n,i(tn) = 0, we can get, for t ∈ [tn, tn+1],

(4.17)∣∣uE

n,i(t)− uHn,i(t)

∣∣ ≤ τh5CH,i(S5n, µ(S5

n), ν(S5n))

for some computable function CH,i. Here we are using the notations µ(S5n) and

ν(S5n) to indicate the fact that µi,q and νi,q (q = 2, 1, 0) can be bounded in terms of

the spatial smoothness indicator. In practice, it is better to compute them directly.By Lemma 4.1,

‖uERn (tn+1, x)− uHR

n (tn+1, x)‖L1(Ω) ≤ τh5CH(S5n),

where CH(S5n) = C1

∑i=1,··· ,M hCH,i(S5

n, µ(S5n), ν(S5

n)). #

For the estimation of the temporal local error, the key is to prove the boundednessof the temporal derivatives of uH

n,i(t) in [tn, tn+1]. To this end, we need the followingnotations. Let

v(l)n,i(t) =

dl

dtluH

n,i(t), l = 0, 1, · · · ,

andv(l)

n (t) = (v(l)n,1(t), · · · , v

(l)n,M (t)).

Lemma 4.9. There are computable functions CV,l such that, for l = 1, · · · , k + 1,∣∣∣∣ dl

dtluH

n,i(t)∣∣∣∣ ≤ ∣∣∣∣ dl

dtluH

n,i(tn)∣∣∣∣+ τCV,l(S5

n, T ln)

for t ∈ [tn, tn+1].

Proof. First, we notice that uHn,i(t) remains bounded as a consequence of (4.17) in

the proof of Lemma 4.8. Based on the definition of uHn,i(t) in (3.2), the Godunov

flux given by (3.3) and the WENO reconstruction (2.9), we know that

duHn,i

dt=

1h

[f

(2∑

r=−2

ηi−1,r uHn,i−1+r

)− f

(2∑

r=−2

ηi,r uHn,i+r

)].

Taking a derivative of this equation with respect to t, we get

dv(1)n,i

dt=

1h

f ′

(2∑

r=−2


)2∑

r=−2

ηi−1,r v(1)n,i−1+r(4.18)

− 1h

f ′

(2∑

r=−2

ηi,r uHn,i+r

)2∑

r=−2

ηi,r v(1)n,i+r.

22 TONG SUN

Let V(1)n (t) = maxi |v(1)

n,i(t)|, we get∣∣∣∣∣dv(1)n,i

dt

∣∣∣∣∣ ≤ β

h

(2∑

r=−2

|ηi−1,r|+2∑

r=−2

|ηi,r|

)V (1)

n (t).

Thus

v(1)n,i(t) ≤ v

(1)n,i(tn) +

∫ t

tn

β

h

(2∑

r=−2

|ηi−1,r|+2∑

r=−2

|ηi,r|

)V (1)

n (s) ds.

Consequently,

V (1)n (t) ≤ V (1)

n (tn) +∫ t

tn

β

h

(2∑

r=−2

|ηi−1,r|+2∑

r=−2

|ηi,r|

)V (1)

n (s) ds.

By the standard Gronwall Lemma,

V (1)n (t) ≤ V (1)

n (tn) exp

(β(t− tn)

h

(2∑

r=−2

|ηi−1,r|+2∑

r=−2

|ηi,r|

)).

From the definition of ηi,r, we always have∑2

r=−2 |ηi,r| < 103 . When the solution is

smooth, we actually have∑2

r=−2 |ηi,r| ≈∑2

r=−2 |ηdr | = 92

60 . So, in general, we canassume that, for some real number σ ∈

(53 , 10

3

),∑2

r=−2 |ηi,r| < σ. In this paper,where we are dealing with smooth solutions, it is safe to choose σ = 1.7. Now thelast inequality can be simplified to

(4.19) |v(1)n,i(t)| ≤ V (1)

n (t) ≤ V (1)n (tn) exp

(2βσ(t− tn)

h

).

We can use (4.19) as a rough estimate of |v(1)n,i(t)|. A fine estimate will be obtained

later. Now

dv(2)n,i

dt=

1h

f ′

(2∑

r=−2


)2∑

r=−2

ηi−1,r v(2)n,i−1+r

− 1h

f ′

(2∑

r=−2

ηi,r uHn,i+r

)2∑

r=−2

ηi,r v(2)n,i+r

+1h

f ′′

(2∑

r=−2


)(2∑

r=−2

ηi−1,r v(1)n,i−1+r

)2

− 1h

f ′′

(2∑

r=−2

ηi,r uHn,i+r

)(2∑

r=−2

ηi,r v(1)n,i+r

)2

.

By taking V(2)n (t) = maxi |v(2)

n,i(t)|, we get

(4.20)

∣∣∣∣∣dv(2)n,i

dt

∣∣∣∣∣ ≤ 2βσ

hV (2)

n (t) +2δσ2

h

[V (1)

n (t)]2

.


By the Simpson’s rule, E(t− tn, βσh ) = 1

6 + 46 exp

(2βσ(t−tn)

h

)+ 1

6 exp(

4βσ(t−tn)h

)> 1

t−tn

∫ t

tnexp

(4βσ(s−tn)

h

)ds. Thus∣∣∣v(2)

n,i(t)∣∣∣ ≤ ∣∣∣v(2)

n,i(tn)∣∣∣+ 2δσ2(t− tn)

hE(t− tn,

βσ

h)[V (1)

n (tn)]2

+∫ t

tn

2βσ

hV (2)

n (s) ds.

Again, by the Gronwall Lemma,

(4.21)∣∣∣v(2)

n,i(t)∣∣∣ ≤ V (2)

n (t) ≤ E1

(t− tn, δ, σ,

β

h

)exp

(2βσ(t− tn)

h

),

where

E1

(t− tn, δ, σ,

β

h

)= V (2)

n (tn) +2δσ2(t− tn)

hE(t− tn,

βσ

h)[V (1)

n (tn)]2

.

With the last estimate on v(2)n,i(t), we can easily refine the estimate on v

(1)n,i(t). Since

d

dtv(1)n,i(t) = v

(2)n,i(t),

we get

(4.22)∣∣∣v(1)

n,i(t)∣∣∣ ≤ ∣∣∣v(1)

n,i(tn)∣∣∣+ ∫ t

tn

E1

(s− tn, δ, σ,

β

h

)exp

(2βσ(s− tn)

h

)ds.

(4.22) is our desired estimate for duHn,i

dt = v(1)n,i. Again, one can use the Simpson’s

rule to compute a very sharp upper bound for the integral in (4.22). The righthand side of (4.22) is indeed bounded by a computable function of S5

n and T 1n of

the form∣∣∣v(1)

n,i(tn)∣∣∣+ (t− tn)CV,1(S5

n, T 1n).

Inductively, we can derive estimates for dluHn,i

dtl = v(l)n,i, l ≥ 2, in a similar way, but

we omit the tedious details. #

Remarks. We make a few remarks to explain the last lemma and the ideas behindits proof.

• If we only need the first and the second derivatives, it might be good enoughto stay with the rough estimates (4.19) and (4.21). However, we do need toestimate the higher order derivatives. When a rough estimate of a low orderderivative enters the estimation of a higher order derivative, the estimatefor the higher order derivative will become too large for practical erroranalysis. The estimate given by (4.22) has improved a lot over estimate(4.19). If desired, one can use (4.22) to improve the right hand side of(4.20), which will consequently further improve (4.21) and (4.22).

• In the computation, one has to take βτh significantly less than 1, otherwise

the WENO-TVDRK solution will show instability. Because of this, thequotients in (4.22) such as 4βστ

h are not large.• The reason that we can get the fine estimate (4.22) is rooted in the semi-

discrete differential equation, which inherited the smoothness maintenanceproperty of the conservation law, shown by Lemma 4.3. When we first de-rived the rough estimate (4.19), smoothness maintenance was not involved.We caught some of the smoothness maintenance property in the rough es-timate (4.21) of v

(2)n,i(t), which enabled us to prove the finer estimate (4.22).

24 TONG SUN

• The solutions of hyperbolic equations only have local dependence, often un-derstood as finite wave speed. However, due to the existence of numericaldiffusion in numerical schemes, the dependence of numerical solutions areno longer purely local. In (4.22), one can see that the dependence is stillmainly local. The part of the estimate reflecting the global dependence isproportional to the time step size τ . There is still some room for improve-ment if we reconsider the spatially global maxima V

(1)n and V

(2)n , but it is

not desired for the case of smooth solutions.• The method used here to establish the smoothness in [tn, tn+1] only works in

the micro time scale (one time step). Nevertheless, it satisfies our needs inthe local error estimation. A macro time scale analysis of numerical smooth-ness maintenance has to take numerical diffusion into account, which is notdone here because it seems too hard. However, the information deliveredby the smoothness indicators is sharper than any possible estimation.

Now we are ready to estimate the local temporal discretization error.

Lemma 4.10. There is a computable function CF , such that

(4.23) ‖uHRn (tn+1, x)− uR

n+1(x)‖L1(Ω) ≤ τk+1CF (S5n, T k

n ).

Proof. With the help of Lemma 4.9, it is a standard local error estimation proce-dure for a k-th order Runge-Kutta method to get the estimate

‖uHn (tn+1)− un+1‖∞ ≤ τk+1CF (S5

n, T kn ),

with a computable function CF . By using Lemma 4.1, we get (4.23). #

Combining Lemma 4.6, Lemma 4.8 and Lemma 4.10, we have the following maintheorem, which shows that the convergence is of optimal order in space and time,the local error in [tn, tn+1] depends on the smoothness of the numerical solution attime tn, and the global error is the simple accumulation of the local error.

Theorem 4.11. Let u(t, x) be the solution of the conservation law (2.1) and uRn (x)

(n = 0, 1, · · · , N) be the reconstructed numerical solution defined in Section 3.1.At the ending time tN = T of the computation, the global error of the numericalsolution is

‖u(tN , x)− uRN (x)‖L1(Ω) ≤ ‖uI − uR

0 ‖L1(Ω)(4.24)

+h5N−1∑n=0

τ[CE(S5

n) + CH(S5n)]+ τk

N−1∑n=0

τCF (S5n, T k

n ),

where uI is the initial solution, uR0 = R(u0,1, · · · , u0,M ), the functions CE(S5

n),CH(S5

n) and CF (S5n, T k

n ) of the smoothness indicators S5n and T k

n are given inLemma 4.6, Lemma 4.8 and Lemma 4.10 respectively.

Proof. For each time step, we split the global error as

‖u(tn+1)− uRn+1‖L1(Ω)

≤ ‖u(tn+1)− uEn (tn+1)‖L1(Ω) + ‖uE

n (tn+1)− uERn (tn+1)‖L1(Ω)

+‖uERn (tn+1)− uHR

n (tn+1)‖L1(Ω) + ‖uHRn (tn+1)− uR

n+1‖L1(Ω)

≤ ‖u(tn)− uRn ‖L1(Ω) + τh5CE(S5

n) + τh5CH(S5n) + τk+1CF (S5

n, T kn ).


Figure 1. M ln (l = 0, · · · , 5), indicating interior smoothness, tn = 1.5

Here we have used the L1-contraction between the entropy solutions and the esti-mates in Lemma 4.6, Lemma 4.8 and Lemma 4.10. The main error estimate (4.24)is proven by repeatedly using the last inequality. #

5. Numerical experiments

The main error estimate given by Theorem 4.11 depends on the boundedness ofthe smoothness indicators. Therefore, to support the error analysis for smooth solu-tions in the previous section, it suffices to show the boundedness of the smoothnessindicators in numerical experiments.

As an example, we have computed the WENO-TVDRK solution of the Burgers’equation

ut + (u2/2)x = 0

in Ω = [0, 10] with the initial condition

uI(0, x) = 0.5 + 0.3 sin(

2πx

10

),

which is perfectly smooth. We take h = 0.05 and τ = 0.005.In Figure 1-4, we have plotted the values of the smoothness indicators at time

t = 1.5. At this time, all the components of the smoothness indicators can clearlybe considered as bounded. Therefore the error estimate of Theorem 4.11 is certainlyapplicable at this time.

While it is a normal procedure to verify that the error analysis is supportedby the computed smoothness indicators for undoubtedly smooth solutions, it ismore interesting to observe the deterioration of the numerical smoothness alongthe sharpening of the solution. In Figure 5-8, we have plotted the values of thesmoothness indicators at time t = 2.5. This is still long before the time of theinitial shock formation at t ≈ 5.305165. However, at this time, some componentsof the smoothness indicators, specifically D4

n and D5n, have started to grow up

significantly.Even though all the spatial and temporal derivatives should blow up simulta-

neously in theory, the higher order derivatives grow up much faster during thesharpening process. This probably means that the higher order accuracy of theWENO schemes does not hold valid for very long during the sharpening toward a

26 TONG SUN

Figure 2. Dln (l = 0, · · · , 5), cell boundary smoothness, tn = 1.5

Figure 3. µq,i and νq,i (q = 0, 1, 2), weights’ smoothness, tn = 1.5

Figure 4. T ln (l = 1, · · · , 4), temporal smoothness, tn = 1.5

shock formation. Careful and detailed numerical experiments are needed to inves-tigate when will the estimate of Theorem 4.11 collapse in the process of |ux| → ∞,and how well the estimate work before the collapse.

6. Conclusion remarks

• The global error estimate given in Theorem 4.11 is the first detailed error estimatefor a WENO scheme on smooth solutions.


Figure 5. M ln (l = 0, · · · , 5), indicating interior smoothness, tn = 2.5

Figure 6. Dln (l = 0, · · · , 5), cell boundary smoothness, tn = 2.5

Figure 7. µq,i and νq,i (q = 0, 1, 2), weights’ smoothness, tn = 2.5

• The numerical experiments support the error analysis, but have also shown that asolution cannot be considered as smooth during its shock formation process, whichactually begins long before the appearance of a shock.• Local error analysis needs to be done separately for at least the following threestages: sharpening processes of solutions toward shock formation, newly formed

28 TONG SUN

Figure 8. T ln (l = 1, · · · , 4), temporal smoothness, tn = 2.5

shocks, and fully developed shocks. However, there is probably no clear cut betweenthe time durations of two neighboring stages.

In the master thesis [9] of Jian Wu, a set of not identical but equivalent smooth-ness indicators have been computed. All the smoothness indicators in [9] and allthe smoothness indicators defined in this paper have been computed throughoutthe development of shocks. The information delivered by these indicators aroundshocks are also very abundant [9]. Since we do not cover the error analysis of non-smooth solutions here, we do not show those numerical results. The author wantsto appreciate Mr. Jian Wu for his participation and contribution in the early stageof this research project.

References

1. B. Cockburn, A simple introduction to error estimation for nonlinear hyperbolic conservation

laws, The Graduate Student’s Guide to Numerical Analysis ’98, Springer, New York, 1999,

pp.1-46.2. G. Jiang and C.-W. Shu, Efficient implementation of weighed ENO schemes, Journal of

Computational Physics, v126 (1996), pp.202-228.

3. C. Johnson and Szepessy, Adaptive finite element methods for conservation laws based on aposteriori error estimates , Comm. Pure Appl. Math. v48 (1995), pp.199-234.

4. Y. LIU, C.-W. Shu and M. Zhang, High order finite difference WENO schemes for nonlinear

degenerate parabolic equations, SIAM J. Sci. Comput., v33 (2011), pp.939-965.5. C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for

hyperbolic conservation laws, Lecture Notes in Mathematics, 1998, Volume 1697/1998, pp.325-

432.6. J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1994.

7. Gilbert Strang, Accurate Partial Difference Methods II: Non-Linear Problems, NumerischeMathematik, v6 (1964), pp.37-46.

8. T. Sun and D. Rumsey, Numerical smoothness and error analysis for RKDG on the scalar

nonlinear conservation laws, submitted to SIAM J. Numer. Anal., September, 2010.9. J. Wu, Numerical smoothness of ENO and WENO schemes for nonlinear conservation laws,

Master’s Thesis, submitted to the Graduate College of Bowling Green State University, Au-

gust, 2011.

Department of Mathematics and Statistics, Bowling Green State University, Bowl-ing Green, OH 43403

E-mail address: [email protected]

NUMERICAL SMOOTHNESS AND ERROR ANALYSIS ON WENO FOR THE NONLINEAR

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