Two-level time-marching scheme using splines for solving the advection equation

*Corresponding author. Tel.: #1-949-824-6126; fax: #1-949-824-8585.

E-mail addresses: [email protected] (K. Nguyen), [email protected] (D. Dabdub).

Atmospheric Environment 35 (2001) 1627}1637

Two-level time-marching scheme using splines for solving theadvection equation

Khoi Nguyen, Donald Dabdub*

Department of Mechanical and Aerospace Engineering, University of California at Irvine, Irvine, CA 92697-3975, USA

Received 17 August 2000; accepted 28 August 2000

Abstract

A new numerical algorithm using quintic splines is developed and analyzed: quintic spline Taylor-series expansion(QSTSE). QSTSE is an Eulerian #ux-based scheme that uses quintic splines to compute space derivatives and Taylorseries expansion to march in time. The new scheme is strictly mass conservative and positive de"nite while maintaininghigh peak retention. The new algorithm is compared against accurate space derivatives (ASD), Galerkin "nite elementtechniques, and the Bott scheme. The cases presented include classical rotational "elds, deformative "elds, as well asa full-scale aerosol model. Research shows that QSTSE presents signi"cant improvements in speed and oscillationsuppression against ASD. Furthermore, QSTSE predicts some of the most accurate results among the schemestested. ( 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Hyperbolic systems; Air quality models; Eulerian methods; Taylor expansion; Finite di!erences

1. Introduction

The classical advection equation often appears in de-velopment of air quality models. The one-dimensionalform of the advection equation is given as

Lc

Lt#

Luc

Lx"0, (1)

where u is the wind velocity and c is the concentra-tion. The extension to higher dimensions in air qualitymodels is usually performed through operator splitting(Yanenko, 1971), which is widely used in air qualitymodeling (Harley et al., 1993). It is well known that thenumerical solution to Eq. (1) leads to dispersions andoscillations (Oran and Boris, 1987). To address this prob-lem, the development of various advection schemes hasbeen a continuing e!ort. A review of such schemes as theyapply to air quality models is found in Rood (1987),

Chock (1985, 1991), Chock and Dunker (1983), and Dab-dub and Seinfeld (1994).

There are numerous techniques used to solve the ad-vection equation. Some of the popular techniques areEulerian "nite di!erences, #ux-based schemes (Bott,1989), Lagrangian characteristics (Augenbaum, 1984)and semi-Lagrangian methods (Purnell, 1976). Eachmethod has merits in certain cases and problems inothers. For example, Lagrangian methods are ideal foradvection processes but incur di$culties since trajecto-ries scatter points non-uniformly. This problem leads toine$ciencies and inaccuracies since data "elds in airquality models are usually given at uniform grid points.Eulerian "nite di!erences are attractive due to their sim-plicity, #ux-based schemes are tailored to be conserva-tive, while semi-Lagrangian methods provide higherstability and thus, larger time steps (Bermejo andStaniforth, 1992). However, semi-Lagrangian methodstend not to obey mass conservation and require addi-tional mass adjustments (Huang, 1996).

This paper presents the development and analysis ofQSTSE, a new Eulerian method to solve the advectionequation. The formulation of the algorithm is presentedin Section 2. The performance tests of the algorithm are

1352-2310/01/$ - see front matter ( 2001 Elsevier Science Ltd. All rights reserved.PII: S 1 3 5 2 - 2 3 1 0 ( 0 0 ) 0 0 4 4 3 - X

presented in Section 3. The performances are gauged byclassical advection tests as well as a three-dimensionalatmospheric aerosol dynamic model.

2. Algorithm description

2.1. The non-yux quintic spline with Taylor-seriesexpansion (NFQSTSE) algorithm

Classical "nite di!erence schemes such as second-or-der central di!erences can be derived from interpolationof data. In particular, second-order central di!erencescan be attained from interpolating piece-wise quadraticpolynomials. Similarly, fourth-order "ve-point "nite dif-ferences can be derived from a quartic polynomial "t. Theuse of splines as an interpolator has become increasinglypopular. Spline interpolators are viable alternatives toLagrange-polynomial interpolators, thus, spline deriva-tive schemes provide competitive alternatives to classicalTaylor "nite di!erence schemes.

In quintic spline interpolation schemes, piece-wisequintic polynomials are "tted to data points requiringcontinuous fourth derivatives globally. Consider a uni-form grid in x with N points, a quintic spline, S

j(x) on the

grid interval (xj, x

j`1) that interpolates nodal values a

jis

Sj(x)"a

j#b

j(x!x

j)#c

j(x!x

j)2#d

j(x!x

j)3

#ej(x!x

j)4#f

j(x!x

j)5, (2)

where 1)j)N!1 and bj, c

j, d

j, e

j, f

jare the linear,

quadratic, cubic, quartic and quintic coe$cients of thespline, respectively. Setting h"x

j`1!x

jand requiring

continuous fourth derivatives, yields the following systemof equations:

aj`1

"aj#b

jh#c

jh2#d

jh3#e

jh4#f

jh5, (3)

bj`1

"bj#2c

jh#3d

jh2#4e

jh3#5f

jh4, (4)

2cj`1

"2cj#6d

jh#12e

jh2#20f

jh3, (5)

6dj`1

"6dj#24e

jh#60f

jh2, (6)

24ej`1

"24ej#120f

jh. (7)

An e$cient solution to these equations is found in DeBoor (1978). Eq. (1) is reduced to an ordinary di!erentialequation in time with the coe$cients of the spline, whichare proportional to the derivatives at nodal points. Toadvance in time, a time-marching scheme (e.g. Runge-Kutta), a Taylor series in time, or even an exponentialsolver can be used. In this paper, a Taylor series in time isused due to its computational e$ciencies. If a truncatedTaylor approximation in time is applied, a fourth-order

expansion or less should be used because the "fth deriva-tive is generally discontinuous with quintic splines. Con-sider a fourth-order Taylor expansion in time expressedas follows:

c(t#*t)+c(t)#4+i/1

Lic

Lti

*ti

i!#h.o.t., (8)

The derivative in time can be substituted by the deriva-tive in space via di!erentiating Eq. (1). Assuming that u istime independent on the interval, dt (which is often ap-plied in air quality models as described by Harley et al.,1993), the second derivative of c with respect to t can beexpressed as

L2c

Lt2"

LLt

Lc

Lt"!

LLx

Luc

Lt"

LLxAu

Luc

Lx B. (9)

Higher-order terms are retrieved using the same principleand are given as

L3c

Lt3"!A

Lu

LxB2Luc

Lx!3u

Lu

Lx

L2uc

Lx2

#uALuc

Lx

L2u

Lx2!u

L3uc

Lx3 B, (10)

L4c

Lt4"A

Lu

LxB3Luc

Lx#7uA

Lu

LxB2L2uc

Lx2#2u

Lu

LxA2Luc

Lx

L2u

Lx2

#3uL2ucLx2 B#u2A4

L2uLx2

L2ucLx2

#

LucLx

L3uLx3

#uL4ucLx4 B.

(11)

With a fourth-order Taylor expansion, it is possible tomarch in time with high accuracy and e$ciency sincederivatives of uc and u with respect to x are readilyavailable in quintic splines. The memory requirement ofthis time scheme is minimal since it is a two-level time-marching scheme or a forward-time technique.

The method described so far, shall be calledNFQSTSE, the non-#ux quintic spline Taylor-series Ex-pansion. The stability and eigenvalue analysis of thismethod is detailed in the appendix.

NFQSTSE is integrally mass conservative. Integralmass conservation is proven simply by integrating Eq. (8)over the entire domain and using splines with periodicend conditions. Periodic splines are commonly used invarious disciplines, where the derivatives at endpoints areequal (Spath, 1995; Dubeau and Savoie, 1983; Spath andMeier, 1988). Starting with Eq. (8)

c(x, t#*t)"c(x, t)#Lc

Lt*t#

L2c

2!Lt2*t2#

L3c

3!Lt3*t3

#

L4c

4!Lt4*t4 (12)

1628 K. Nguyen, D. Dabdub / Atmospheric Environment 35 (2001) 1627}1637

and applying Eq. (1) yields

c(x, t#*t)"c(x, t)!Luc

Lx*t#

LLxCu

Luc

Lx D*t2

2!

!

LLxCu

LLxCu

Luc

Lx DD*t3

3!

#

LLxCu

LLxCu

LLxCu

Luc

Lx DDD*t4

4!. (13)

Integrating both side over the domain would producea mass balance,

Pxf

x0

c(x, t#*t) dx"Pxf

x0

c(x, t) dx!Pxf

x0

Luc

Lx*t

#

LLxCu

Luc

Lx D*t2

2!

!

LLxCu

LLxCu

Luc

Lx DD*t3

3!

#

LLxCu

LLxCu

LLxCu

Luc

Lx DDD*t4

4!dx,

(14)

where [x0,x

f] is the interval of the domain. If the expres-

sions inside the integral is continuous or de"ned uniquelyover the domain and with speci"ed periodic conditions,uc(x),u(x)c(x), uc(x

0)"uc(x

f), Liuc(x

0)/Lxi"Liuc(x

f)/

Lxi for i"124 then Eq. (14) reduces to the conservationof mass in integral form

Pxf

x0

c(x, t#*t) dx"Pxf

x0

c(x, t) dx. (15)

2.2. Quintic spline with Taylor-series expansion (QSTSE)algorithm

QSTSE is a discretely mass conservative #ux-basedscheme. It is derived from NFQSTSE by similar fashionas the proof of NFQSTSE's integral mass conservation.If Eq. (8) is evaluated at cell centers, the integration overeach cell yields a #ux system,

Pxj`1

xj

c(x, t#*t) dx"Pxj`1

xj

c(x, t) dx#*tPxj`1

xj

Lc

Lt

#

L2c

2!Lt2*t#

L3c

3!Lt3*t2

#

L4c

4!Lt4*t3dx, (16)

where xj

and xj`1

is the position of the cell's left andright boundaries, respectively. Integrating Eq. (16) yields,

Cn`1j

*x"Cnj*x#*t(F

j`1!F

j). (17)

Here, Cn`1j

, Cnj, F

j`1and F

jare the average concentra-

tion at time step n#1, average concentration at timestep n, #ux from the right boundary and #ux from the leftboundary respectively. The sum of Eq. (17) over j is thestatement of discrete mass conservation. Depending onhow the integration of Eq. (16) is performed, di!erentdiscretely conservative schemes arise. In particular, if theintegration is performed analytically, the #ux is given as

Fj"uc(x

j)#Cu(x

j)Luc(x

j)

Lx D*t

2!

#Cu(xj)

LLxCu(x

j)Luc(x

j)

Lx DD*t2

3!

#Cu(xj)

LLxCu(x

j)

LLxCu(x

j)Luc(x

j)

Lx DDD*t3

4!. (18)

where the derivatives are evaluated by quintic splines.However, the performance of this method is di!usivecompared to NFQSTSE described in Section 2.1. Toremedy this di!usive problem, QSTSE uses a histospline(area-preserving spline) to compute the integration(Schoenberg, 1983; Spath, 1995). By using a histospline,QSTSE maintains the high accuracy of NFQSTSE but isdiscretely mass conservative since it is a #ux formulation.

QSTSE is positive de"nite. Namely, positive de"nite-ness is maintained by #ux limitation similar to that usedby Bott (1989). If the total out#ow is limited by howmuch mass is available. Mathematically that is given as

Fj`1

*!

*x

*tCn

j#F

j. (19)

Splines are well suited for many interpolation prob-lems like that in advection. In this paper, area preserving,periodic and natural splines are implemented. Manyspline interpolators exist that are positive de"nite(Schmidt and Hess, 1988; Fischer et al., 1991), shapepreserving (Costantini, 1988; McAllister et al., 1977), andmonotonic (Costantini, 1986). These spline properties areideal for #ux-integrated semi-Lagrangian methods, but isnot required in the approach developed here since theinterpolation is performed on the #ux and not the con-centration.

3. Description of numerical tests

Test cases are chosen to gauge the performanceof QSTSE under a variety of conditions. The rotatingcosine hill in test I, as proposed by Pepper and Long(1978), provides fundamental requirements for advectionschemes: peak retention. The cosine hill distribution ischosen since it represents typical pro"les of high pollu-tant concentration while maintaining a stringent testcase. In test II, a rugged concentration pro"le is advected

K. Nguyen, D. Dabdub / Atmospheric Environment 35 (2001) 1627}1637 1629

Table 1Performance of various un"ltered schemes on the cosine hill test after two revolutions. MI, MC, MD, and ME are the minimumconcentration ratio, mass conservation ratio, mass distribution ratio, and maximum error ratio, respectively

Algorithm Reference Description Peak MI MC MD ME Rel. Time

GLK Dabdub and Seinfeld (1994) Fourth-order Taylor}Galerkin 91 !0.14 0.999 0.999 0.283 1.0ASD Gazdag (1973) Taylor expansion with Fourier

derivatives98 !0.01 0.999 0.981 0.132 40.7

QSTSE This work Fourth-order Taylor expansionwith quintic splines

101 !0.01 1.000 0.998 0.021 5.8

NFQSTSE2 This work Second-order Taylor expansionwith quintic splines

104 !0.18 1.002 1.073 0.341 5.2

NFCSTSE This work Second-order Taylor expansion withcubic splines

78 !0.04 1.000 1.930 0.213 3.2

BOTT4 Bott (1988) Nonlinear #ux normalization #uxscheme

77 !0.10 1.000 1.125 0.237 2.3

to gauge the performance of QSTSE under stringentmodulation. Conservation of mass under divergent #ow"elds is measured in test III where a sharp spike isadvected through a velocity ramp. The #ow of Smolar-kiewicz (1982) tests QSTSE under conditions that aremassively deformative in test IV. Though this conditionis more stringent than actual atmospheric conditions, itprovides a good indicator of the scheme's ability toevolve correctly under a strongly deformative "eld. TestV gauges the behavior of the QSTSE under more realisticatmospheric conditions of the South Coast Air Basinof California. Test V implements QSTSE into a fullthree-dimensional grid-based air quality model, the CITAirshed Model (Harley et al., 1993).

The schemes that are selected to be compared withQSTSE are summarized in Table 1. NFCSTSE isa Taylor expansion in time using a cubic spline anda second-order marching in time. NFQSTSE2 is similarto NFQSTSE but uses a second-order marching in timeinstead of a fourth order. GLK is chosen for its fast speed.ASD (Gazdag, 1973) is used for being most accuratebeyond some 20 other solvers as presented by Chock(1991, 1985), Chock and Dunker (1983) and con"rmed byDabdub and Seinfeld (1994). Bott solvers present com-petitive accuracy, speed and mathematical conservationproperties (Bott, 1989; Dhaniyala and Wexler, 1996).

Test I: Rotating cosine hill: The test consists of a 33]33uniform mesh with center at (17, 17) and a spacing of 1.The test is described as

Lc

Lt#ux

Lc

Ly!uy

Lc

Lx"0,

where u is such that the cosine hill rotates two revol-utions in 7200p time units. The test has the followinginitial and boundary conditions

c"10 on the boundary,

c(x, y)"G45[1#cos(rp

4)]#10 if r)4,

10 otherwise,

where r"J(x!7)2#(y!17)2. The time incrementused in this test is 30p, thus, requiring 240 time steps tocomplete two revolutions.

Test II: Rugged proxle: Suitability of QSTSE with rug-ged concentration pro"le undergoing advection is testedhere. Consider a one-dimensional grid with 33 pointswith space increments of 1.0. The advective wind "eld isa constant wind "eld with Courant number of 0.2. A rug-ged concentration pro"le with thin square waves of vari-ous distances apart is transported downwind.Mathematically, the case is given as

u(x)"0.2 (20)

with two initial conditions representing various spacingsare

ci(x)"G

1 if x"2, 3, 4#i, 5#i,

0 otherwise

for i"1 and i"2. These initial conditions are advecteddownwind for 20 grid points or 100 time steps.

Test III: A ramp velocity proxle: The following test ispresented in Chock et al. (1996) and consist of a singlespike advecting through a divergent "eld. Consider a gridof 33 points with the following initial condition given by:

c(x)"G1 if x"5,

0 otherwise

with a velocity pro"le of

u(x)"G0.1 if x)10,

0.1#(x~10)50

if 10)x)15,

0.2 if x*15.

The test gauges QSTSE's mass conservation perfor-mance as it goes through the velocity ramp with a timestep of 1 for 125 time steps.


Test IV: Deformative wind xeld: Consider a 100]100uniform grid in two dimensions with velocity pro"les

u(x, y)"8p

25sinA

px

25BsinApy

25B (21)

v(x, y)"8p

25cosA

px

25B cosApy

25B. (22)

With initial conditions,

c(x, y)"G100!100r

15if r)15,

0 otherwise,

where r"J(x!5)2#(y!50)2. The space increment isunity and the time increment, *t"0.7. The exact graphi-cal solution to this test is presented by Staniforth et al.(1987) and the solution should be compared to the exactsolution after short integration time (Bott, 1989).

Test V: Practical application: The "nal test is a fullimplementation in the CIT airshed model (Harley et al.,1993). The CIT model includes advection, di!usion,chemistry, deposition and emissions. The episode simu-lates 27 August 1987 for the South Coast Air Basin ofCalifornia. The domain is irregular and has 5 layers inthe vertical direction with 994 grid points in each layer.The spacing between each grid point in a layer is 5000 m.There are 35 species in the gas phase, this paper focuseson ozone concentration. The wind "elds are interpolatedfrom observed data via the method described in Goodinet al. (1979) with a divergence criterion to be less than10~3 s~1. In this model run, Forester "lters (Forester,1977) were used to suppress negative mass produced byASD and GLK. No "lters are used with QSTSE.

4. Results

To measure the relative accuracy among the algo-rithms, the following performance indices are evaluated:

MC"

+x,y

c(x, y, t)

+x,y

c(x, y,0), (23)

MD"

+x,y

c(x, y, t)2

+x,y

c(x, y,0)2, (24)

MA"

Max[c(x,y, t)]

100, (25)

MI"Min[c(x,y, t)!10.0]

100, (26)

ME"

MaxDc(x,y, t)!c%9!#5

(x, y, t)D100

. (27)

MC, MD, MA, MI, and ME are the mass conservationratio, mass distribution ratio, maximum concentration

ratio, minimum concentration ratio, and maximum errorratio, respectively.

Fig. 1 shows the result of QSTSE for test case I, therotating cosine hill. After two revolutions the maximumpeak of 100 is maintained closely with QSTSE. Table 1summarizes the results of the schemes after two revol-utions of the cosine hill. ASD and QSTSE retain theirmaximum peak values within 5%, but other schemesdeteriorate. Concentrations that are below the back-ground level of 10 is kept to a minimum with ASD,QSTSE, and NFCSTSE methods, but is substantial withthe GLK, NFQSTSE2, and BOTT4. Mass conservationis well preserved for all schemes with QSTSE andBOTT4 preserving mass conservation exactly.

The computational times for these schemes vary sub-stantially and are reported in Table 1. Normalizing theGLK scheme to consume 1 time unit, QSTSE, BOTT4and ASD consume 5.8, 2.3 and 40.7 time units, respec-tively. These times are computed on a sequential machinewith compiler optimization. The computational cost ofQSTSE is not substantially more than that of NFQSTSEor NFQSTSE2, but has better performance character-istics than both of these low-order schemes.

The results of test II are presented in Figs. 2 and 3.Fig. 2 shows the performance of the schemes under rug-ged conditions with no spacing between the two squarewaves. Exact mass conservation and positive de"nitenessis achieved by both QSTSE and BOTT4. ASD and GLKaccrued 2 and 8% mass, respectively. ASD and QSTSEare able to resolve some separation between the twosquare waves, while BOTT4 and GLK lumped the twosquare waves together. ASD overshoots the height whileall other schemes underpredicts the height in this test.With a spacing of one grid point between the squarewaves, the various schemes perform better than withoutany spacing as shown in Fig. 3. QSTSE and BOTT4 arepositive de"nite and strictly mass conservative while theothers are not. With the increased spacing, all schemesexhibit some separation between the two square waves asit advects downwind. The best separation is maintainedby ASD but has some serious under and over predictions.QSTSE is able to resolve the separation in addition toa height increase closer to the exact solution. Furthertests conducted (not shown) with more spacing betweenthe two square waves show performance increase in allschemes as the spacing increases.

Test III is designed primarily to gauge mass conserva-tion among the schemes as a sharp spike passes througha divergent wind "eld. Fig. 4 presents the results fromthat test. The exact solution is derived from a Lagrangianapproach. In essence, the spike located at grid point5 broadens when it passes a velocity ramp. The exactsolution is a spike near grid point 23 with a height of 0.5and a base of radius 2. Both QSTSE and BOTT4 areexactly mass conservative and positive de"nite. However,ASD and GLK accrued 36% and 6% mass, respectively.


Fig. 1. Concentration pro"le of the rotating cosine hill advected by QSTSE for two revolutions at *t"30p. The peak and form of thecosine hill is preserved almost identically. The 20 contours are equally spaced contours from !2 to 100.

Fig. 2. Concentration pro"le of Test II with rugged concentration pro"le. There is no grid spacing between the two square waves as it isadvected downwind.


Fig. 3. Similar to Fig. 2 with the two square waves separated with one grid spacing. As the spacing increase all schemes perform better.

Fig. 4. Concentration pro"le resulting from test IV, a strenuous case of advecting a spike down a velocity ramp. The purpose of this testis measuring mass conservation.


Fig. 5. Concentration pro"le as computed by QSTSE for the deformative #ow of Smolarkiewicz (1982) after 19, 38, 57, and 75 iterations,respectively. Contours are shown at concentrations of 1, 10, 20, 30, 40, 50, 60, 70, 80, 90 and 100.

ASD predicts the peak most accurately but su!ers fromsubstantial oscillations. QSTSE, GLK and BOTT4underpredict the peak but QSTSE retains the peak andits location better than GLK and BOTT4.

QSTSE concentration pro"les for test case IV areshown in Fig. 5. The exact solution to this problem ispresented in Staniforth et al. (1987). Comparing thegraphs of the exact solution to the numerical one showsQSTSE's ability to retain both peak and shape undera deformative #ow. Furthermore, Fig. 5 and numericaldata show QSTSE's ability to preserve symmetryover the reported time increments as performed alsoby Bott (1989). The maximum error occured at thepeak where the results of QSTSE reports 82.51 insteadof 100.

Results from the full model implementation are shownin Fig. 6. The model run was made to compare QSTSE,ASD, and GLK. ASD is selected because of its accuracyand GLK its wide use and e$ciency. Fig. 6 showsthe measured ozone concentrations for the city of

Claremont, a location with high concentration of ozone.All schemes present similar qualitative results for ozoneconcentrations at Claremont. Ozone contours through-out all the simulation hours con"rm this similarity be-tween the three schemes. Results produced by ASD andQSTSE are almost identical and are closer to the ob-served data than results produced by GLK. In general,GLK tends to underestimate ozone concentration duringthe late afternoon. As described previously, the timerequirements among the solvers di!ers by a factor of 40in pure advection test cases. However, in a full model runthe overall computational time when using di!erentschemes di!ers only by a factor of 4 since the chemistrycomputations consume most of the CPU time. In par-ticular, the time required by GLK and QSTSE is approx-imately equal. On the other hand, a full model runthe ASD is approximately 4 times slower. Given thatGLK took 1 time unit to complete the full model run,QSTSE and ASD consumed 1.02 and 3.95 time units,respectively.


Fig. 6. Ozone concentrations at the city of Claremont as simulated by the CIT Airshed Model for 27 and 28 August of 1987. Both ASDand QSTSE predict less di!usive ozone concentrations that are closer to the observed data than those predicted by GLK.

Fig. 7. Eigenvalue distribution of QSTSE for 30 grid points at various CFL numbers. At lower CFL numbers the eigenvaluedistribution indicates excellent amplitude and phase preservation.


5. Conclusion

The advection equation plays a dominant role in airquality models. The inherent non-linearity of the chem-istry operator emphasizes the importance of an accurateadvection solver (Ho( v et al., 1989). One of the mostaccurate advection solvers for air quality models re-ported in the literature is the accurate space derivative(ASD) method (Chock, 1991; Dabdub and Seinfeld,1994). However, it is computational expensive and re-quires extraneous periodic boundary conditions. In thispaper, quintic spline interpolation techniques are used tosolve the advection equation. The new scheme, QSTSE,is signi"cantly faster and produces similar peak retentionproperties than ASD in classical test cases while main-taining both positive de"niteness and exact mass conser-vation. In a full model run, ASD and QSTSE producenearly identical results. QSTSE is slightly slower thanthe Taylor}Galrerkin scheme (currently implemented),Taylor}Galerkin (GLK). However, the increased accu-racy of QSTSE over GLK improves predicted peakozone concentrations in a full model simulation by27.5%.

Acknowledgements

This project has been funded in part with FederalFunds from Environmental Protection Agency undera Grant No. R 826371-01-0 and the Earth SystemsScience thrust area of the National Partnership forAdvanced Computational Infrastructure. The results andcontent of this publication do not re#ect necessarily,the views and opinions of the Agency or the USGovernment.

Appendix A

The classical eigenvalue analysis on stability by thematrix method (Hirsch, 1990) provides useful informa-tion of the behavior the NFQSTSE. The eigenvalue stab-ility of the NFQSTSE is applied on the model equation(Hirsch, 1990),

Lc

Lt"!c

Lc

Lx, (A.1)

where c is a constant speed.In order to apply the matrix stability method, the

solution to Eqs. (3)}(7) should be cast in matrix form

b"Ba, (A.2)

c"Ca

2, (A.3)

d"Da

6, (A.4)

e"Ea

24, (A.5)

f"Fa

120, (A.6)

where B, C, D, E, F are the matrix operators that com-putes the coe$cients of the quintic spline. With thecoe$cients represented in matrix form, the advancementof the solution to Eq. (A.1) by the algorithm described inSection 2.1 is given as

cn`1"AI!*tcB#

*t2

2!c2C!

*t3

3!c3D#

*t4c4E4! Bcn,

(A.7)

where cn is the nodal value of the concentration at timestep n. The eigenvalues of the matrix,

S"AI!*tcB#

*t2

2!c2C!

*t3

3!c3D#

*t4c4E4! B, (A.8)

detail the stability behavior of NFQSTSE. The computa-tion of the full matrix, S, depends on the number of gridpoints N and, thus, making the analytical eigenvalues ofS elusive. Instead, the eigenvalues are obtained numer-ically for a wide range of CFL"c*t/*x numbers. StableCFL numbers are extrapolated by analyzing the graph ofthe spectral radius, o(S), where o(S))1. From the resultsobtained, the CFL stability criterion for the NFQSTSE issatis"ed providing that

CF¸(1.0. (A.9)

Fig. 7 presents eigenvalue distributions of the NFQSTSEwith N"30 points for CFL"0.1, 0.3, and 0.5. At lowerCFL numbers the NFQSTSE is able to transport thesignal e!ectively without much damping or dispersion asindicated with the eigenvalues near 1.0. As CFL increasesthe eigenvalues spread out inside the stability circle and isexpected to dampen the signal, as do most solvers.

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Two-level time-marching scheme using splines for solving the advection equation

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