A general technique for assessing the numerical accuracyof solute transport models

Feng RuanAdvanced Technology, Schlumberger, GeoQuest, Houston, Texas

Dennis McLaughlinDepartment of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge

Shuguang LiDepartment of Civil Engineering, Portland State University, Portland, Oregon

Abstract. This note describes a procedure for evaluating the accuracy of numerical solutetransport models in situations where exact closed-form solutions are difficult or evenimpossible to obtain. The procedure attempts to match a specified closed-form “testsolution” by adding forcing terms to the original equation, which is solved numerically.The quality of the match provides valuable information about the performance of thenumerical algorithm. We illustrate this “prescribed forcing method” with an examplewhich simulates solute transport in a heterogeneous velocity field. The numerical solverconsidered in the example is based on the Eulerian-Lagrangian method with linearvelocity and concentration interpolation. Two test solutions of different degrees ofdifficulty are considered. Differences between the exact and numerical test solutions forthe example clearly reveal the influence of grid resolution on model accuracy. Theexample demonstrates that the prescribed forcing method can be used to assess numericalaccuracy in practical situations where model inputs are highly variable and the truesolution is unknown.

1. Introduction

Numerical solutions to partial differential equations are typ-ically checked by comparing them to closed-form (or analyti-cal) solutions. If the closed-form solution is evaluated cor-rectly, any discrepancies between the numerical and closed-form results can be attributed to errors in the numericalalgorithm. Although such checks are useful, they test the nu-merical algorithm only under the simplified conditions neededto obtain a closed-form solution. In solute transport applica-tions the closed-form solutions used for testing purposes typ-ically apply for spatially uniform and time-invariant velocityfields, simple source geometries, and semi-infinite or infinitesolution domains [Javandel et al., 1984]. Numerical algorithmswhich perform adequately under such conditions may not do aswell when the velocity field is highly variable, the source ge-ometry is irregular, or the boundaries are complex.

In order to properly evaluate the performance of numericalalgorithms, we need to go beyond comparisons with simpleclosed-form solutions. A number of different options are rou-tinely used in solute transport applications. One is to check theability of the numerical algorithm to conserve mass in thepresence of a nonuniform velocity field, irregular boundaries,etc. While mass conservation is an important aspect of algo-rithm performance, it does not insure that the numerical so-lution actually satisfies the original transport equation. Massconservative algorithms can, for example, still introduce signif-

icant numerical dispersion. Additional insight can sometimesbe obtained by comparing alternative numerical solutions to a“benchmark problem” which cannot be solved in closed form[Baptista et al., 1995]. Although this provides informationabout the relative performance of the selected algorithms, itdoes not test absolute accuracy. Another option is to examinethe behavior of the numerical solution as its spatial computa-tional grid and time step are progressively refined. If the nu-merical algorithm is working properly, the resulting set ofsolutions should gradually converge to a stable solution as thegrid spacing and time step are progressively reduced. Thisapproach is useful, but it can be computationally demanding.

Some of the deficiencies of traditional methods for testingnumerical algorithms are addressed by the “prescribed forcingmethod,” originally proposed by Dee [1991]. This procedureattempts to match a specified closed-form “test solution” byadding forcing terms to the original equation, which is solvednumerically. The quality of the match provides valuable infor-mation about the performance of the numerical algorithm. Theprescribed forcing method has been used primarily to evaluatemultidimensional hydrodynamic flow solvers [Dee, 1991; Di-etrich et al., 1990; Dee et al., 1992; Roach, 1994, 1995]. In thisnote we generalize the prescribed forcing concept to include awider range of test solutions. We also use the method toinvestigate the accuracy of a popular solute transport solver.

The particular problem of interest here is one where thevelocity field is time invariant but highly variable in space (i.e.,the velocity is a stationary random field with a correlation scalemuch smaller than the size of the solution domain). This is thesituation often considered in theoretical, experimental, andnumerical studies of macrodispersion [Gelhar, 1993]. Since

Copyright 1999 by the American Geophysical Union.

Paper number 1999WR900244.0043-1397/99/1999WR900244$09.00

WATER RESOURCES RESEARCH, VOL. 35, NO. 12, PAGES 3961–3966, DECEMBER 1999

3961

heterogeneous velocity transport problems are difficult tosolve, some care must be taken when using numerical modelsto verify theoretical predictions or analyze field experiments.Inaccurate solvers which are prone to numerical dispersion oroscillation may yield misleading macrodispersivity estimates ormay be reliable only when used with unrealistically large localdispersivity values. The prescribed forcing method provides auseful way to check the adequacy of the grid resolution andtime step used to solve transport problems with heterogeneousvelocity fields. More generally, the method may be used toidentify coding errors, compare alternative solution tech-niques, and provide quantitative performance measures for awide range of applications [Ruan, 1997].

2. MethodThe prescribed forcing method can be applied to many types

of partial differential equations [Dee, 1991; Dietrich et al., 1990;Dee et al., 1992]. For specificity, we focus here on the followingmultidimensional solute transport (advection-dispersion) equa-tion, defined over the spatial domain V and time period J:

­c­t 1 ¹ z ~vc! 2 ¹ z ~D¹c! 5 s ~x , t! [ V 3 J

c~x , 0! 5 c0~x! x [ V (1)

ac 1 b¹c z n 5 g~x , t! ~x , t! [ G 3 J ,

where c(x, t) is the solute concentration, v(x) is the time-invariant velocity, D is a constant scalar dispersion coefficient,s is an interior source/sink, G is the boundary with outward unitnormal n, a, b, and g(x, t) are coefficients used to determinethe type of boundary condition applied, and c0(x) is the initialcondition.

The generalized prescribed forcing method used here mod-ifies the original equation by adding the specified (or pre-scribed) forcing terms s9(x, t), c90(x), and g9(x, t) to thesource, initial condition, and boundary condition. The result-ing modified equation has a solution c*(x, t) (called the testsolution), which will be equal to c(x, t) when the prescribedforcing terms are all zero but will generally be different fromc(x, t) when any of these terms is nonzero. The modifiedequation has the following form:

­c*­t 1 ¹ z ~vc*! 2 ¹ z ~D¹c*! 5 s 1 s9 ~x , t! [ V 3 J

c*~x , 0! 5 c0~x! 1 c90~x! x [ V (2)

ac* 1 b¹c* z n 5 g~x , t! 1 g9~x , t! ~x , t! [ G 3 J .

In principle, a specified differentiable test solution c* willsatisfy (2) if the prescribed forcing terms are computed asfollows:

s9~x , t! 5­c*­t 1 ¹ z ~vc*! 2 ¹ z ~D¹c*! 2 s

~x , t! [ V 3 J

c90~x! 5 c*~x , 0! 2 c0~x! x [ V

(3)

g9~x , t! 5 ac* 1 b¹c z n 2 g~x , t! ~x , t! [ G 3 J .

When the prescribed forcings are selected in this way, thenumerical solution to (2), which we represent by c(x, t),

should be equal to c*(x, t). In practice, these two solutionswill be different because the numerical algorithm is not per-fect. The differences between c(x, t) and c*(x, t) providevaluable quantitative information about the performance ofthe algorithm and about possible sources of numerical error.For example, we can measure the aggregate “goodness of fit”for a two-dimensional problem with the following spatiallyaggregated root-mean-square error measure:

J~t! 5Dx1Dx2

M~t! H Ok51

N

@c*~xk, t! 2 c~xk, t!#2J 1/ 2

, (4)

where M(t) is the plume mass (practically constant in test prob-lems considered here), the subscript k refers to a particular nodeof the numerical grid, N is the total node number of grid nodes,and Dx1 and Dx2 are the grid cell dimensions (node spacings) inthe x1 and x2 directions, respectively. It is easy to think of otherquantitative measures of difference between c* and c, such as thecumulative distribution of errors (ranked from smallest to larg-est). The root-mean-square measure provides a good overall in-dication of the quality of the numerical solution.

It seems reasonable to expect that some test solutions will bebetter suited to the purpose of identifying numerical errorsthan others. We can state three general criteria which will besatisfied by a “good” test solution:

1. The differentiable test solution should be expressed in aclosed form which can be evaluated with essentially no error.Moreover, it should be possible to evaluate the prescribedforcings analytically. This eliminates the possibility that dis-crepancies between the numerical and test solutions are due toerrors in evaluation of the test solution or in computation ofthe prescribed forcing terms.

2. The prescribed forcing terms should not be much largerthan the largest term in the original differential equation (asmeasured by an appropriate norm). This insures that the testchecks the algorithm’s ability to solve the original problem,rather than a much different problem which is dominated bythe prescribed forcings. This criterion is satisfied in the solutetransport problem considered here if the shape and trajectoryof the test solution plume (as measured by first and secondspatial moments) are similar to those of the original solution.

3. The test solution should challenge the numerical solver.For example, the space and time gradients should be suffi-ciently large to insure that the fit between test and numericalsolution becomes noticeably worse when the space and timesteps are increased beyond a certain point. This makes it easierto discriminate between alternative numerical solution algo-rithms, grid resolutions, etc.

In summary, the art of using the prescribed forcing methodis to find a test solution which can be expressed in closed formbut which is sufficiently complex to reveal the deficiencies ofthe numerical algorithm.

3. ExamplesIn this note we evaluate a classical Eulerian-Lagrangian

transport solver for a two-dimensional problem with a randomvelocity field, a simple localized instantaneous source (or initialcondition), and boundaries sufficiently far removed from thesource for V to be considered an infinite domain. The Eule-rian-Lagrangian solution algorithm is based on operator-splitting concepts and relies on linear interpolation of the ve-

RUAN ET AL.: TECHNICAL NOTE3962

locity and concentration fields in the advective step and a finitedifference approximation in the dispersive step [Cheng et al., 1984;Ruan and McLaughlin, 1999; Wheeler and Dawson, 1988]. Wefocus on the effect of spatial grid resolution on the numerical

accuracy of the algorithm. The computational grid consists ofsquare cells (Dx 5 Dx1 5 Dx2) and extends over a rectangularregion of 480 by 240 m. (see Plate 1). The solution is computedover a 2400 day period, with 120 time steps of 20.0 days each. The

Plate 1. Solute contours obtained (first or left column) for the original problem, (second column) for theGaussian test solution, and (third column) for the sinusoidal Gaussian test solution. The four rows, arrangedfrom top to bottom, are obtained with grid spacings of Dx 5 2.0 m, 1.0 m, 0.5 m, and 0.25 m, respectively.The initial center of mass location is at x0 5 48.0 m and y0 5 60.0 m. The contour axes units are meters.

3963RUAN ET AL.: TECHNICAL NOTE

lowest resolution grid we consider has 241 by 121 nodes and a gridcell size Dx of 2 m. The more refined grids have progressivelysmaller grid spacings and larger numbers of nodes.

The random velocity field for the coarsest grid problem isobtained from a spectrally based random field generator [Ruanand McLaughlin, 1998]. Its spatial structure is defined by ve-locity spectra derived from a linearized analysis of the ground-water flow equation and Darcy’s law and an assumed log hy-draulic conductivity spectral density function. The meanvelocity v# is aligned with the x1 direction and has a magnitudeof 0.042 m/d. The log hydraulic conductivity has a Gaussianspectral density with a variance of 1.0. The isotropic correla-tion length is 4Dx or 8 m. The spatial discretization process canbe viewed in the spectral domain as a low-pass filter whichtends to attenuate velocity fluctuations at wave numbers abovea cutoff of approximately 2/Dx [Papoulis, 1984]. Grids withdifferent node spacings will have different cutoffs, and theresulting filtered velocity fields will have different spectralproperties. In order to eliminate this effect, the velocity fieldsfor the higher-resolution grids are obtained by interpolatingthe coarsest resolution field with a cubic spline. This insuresthat the velocity spectrum cutoff will always be approximatelyequal to the value associated with the coarsest grid.

The initial condition for the example problem is

c0~x! 5 cm exp F2~ x1 2 x10!

2

2s102 2

~ x2 2 x20!2

2s202 G . (5)

This defines a spatially distributed instantaneous source cen-tered at [ x10, x20] with characteristic length scales s10 and s20.For our example, x10 5 48.0 m, x20 5 60.0 m, s10 5 12 m,s20 5 4 m, and the initial peak concentration at [ x10, x20] iscm 5 1 ppm. The isotropic scalar dispersion coefficient is0.001 m2/d.

The example problem defined above must be solved numer-ically because the velocity field is random. We use two differ-ent test solutions to evaluate the performance of the Eulerian-Lagrangian numerical algorithm. The first of these is thesolution to a transport equation which has the same initialcondition, infinite boundaries, and constant dispersion coeffi-cient as the example problem but with a spatially uniformmean velocity v# . In this case the test solution c*G(x, t) can beexpressed in closed form as

c*G~x, t! 5 cm

s10s20

s1~t!s2~t!expF2

~x1 2 x10 2 v#t!2

2s12~t!

2~x2 2 x20!

2

2s22~t! G,

(6)

where

s12~t! 5 s10

2 1 2Dt

s22~t! 5 s20

2 1 2Dt .

We call this the “Gaussian test solution.” Note that this solu-tion is derived from a transport equation only because thishelps to satisfy criterion 2 above. When the velocity variance issmall, the test solution concentration contours will have ap-proximately the same shape and overall dimensions as thoseobtained for the random velocity case. Qualitative and quan-titative differences between the concentration contours for thetest and random velocity plumes generally increase as the ve-locity variance increases.

The prescribed forcing terms needed to generate the Gauss-

ian test solution are obtained by substituting (6) into (3). Theresult is

s9G~x , t! 5 ~v1 2 v# !­c*G­ x1

1 v2

­c*G­ x2

~x , t! [ V 3 J

c90G~x! 5 0 x [ V (7)

g9G~x , t! 5 0 ~x , t! [ G 3 J .

The prescribed forcing terms for the initial and boundary con-dition are zero because the selected test solution satisfies theinitial and boundary conditions of the original random velocityproblem. In general, these terms will not be zero. Note that theproblem which results when (7) is substituted into (2) has bothspatially variable velocity and spatially variable prescribed forc-ing terms. If the numerical algorithm is accurate, the effects ofthese terms must cancel in such a way as to produce the desiredGaussian test solution.

The second test solution we consider is not the solution to atransport equation but is obtained by modifying the Gaussiantest solution in a way which increases the difficulty of thenumerical problem. In particular, we multiply the Gaussiantest solution by a harmonic term which induces more spatialvariability in both the test solution and the prescribed forcingterms:

c*SG~x , t! 5 c*H~ x1, t!c*G~ x1, x2, t!

5 A sin @2pf~ x1 2 x10 2 v# t! 1 B#c*G~ x1, x2, t! ,

(8)

where c*H( x1, t) 5 A sin[2pf( x1 2 x10 2 v# t) 1 B] and thespatial frequency f 5 k/s10 is a function of the initial longi-tudinal plume dimension s10. The coefficients k , A , and B arethe fluctuation length, amplitude, and phase coefficients, re-spectively. We call c*SG the “sinusoidal Gaussian test solution.”For the example considered here, k 5 1.0, A 5 0.25, andB 5 0.83. These coefficients were selected to keep the spatialgradients of the sinusoidal Gaussian test solution small enoughto fulfill criterion 2 of section 2, but they are large enough tomake this solution more challenging than the smoother Gauss-ian solution. In addition, setting k equal to 1.0 insures that thewavelength of the harmonic term is larger than the coarsestgrid spacing, so that the prescribed forcing fluctuations areproperly resolved.

The prescribed forcing terms needed to generate the sinu-soidal Gaussian test solution are obtained by substituting (8)into (3). The result is

s9SG~x , t! 5 c*HF ~v1 2 v# !­c*G­ x1

1 v2

­c*G­ x2

G1 c*GF ­c*H

­t 1 v#­c*H­ x1

2­

­ x1D

­c*H­ x1

G2 2D

­c*G­ x1

­c*H­ x1

~x , t! [ V 3 J

c90SG~x! 5 c0~x!$A sin @2pf~ x1 2 x10! 1 B# 2 1%

x [ V g9SG~x , t! 5 0 ~x , t! [ G 3 J .(9)

In this case the prescribed initial condition forcing term is nolonger zero since the sinusoidal test solution does not satisfythe initial condition of the original problem. However, the test

RUAN ET AL.: TECHNICAL NOTE3964

solution still satisfies the original infinite domain boundarycondition, so the prescribed boundary condition forcing term iszero. The various spatial and temporal derivatives appearing in(9) can be computed in closed form from the harmonic andGaussian portions of the sinusoidal Gaussian test solutiongiven in (8).

The accuracy of the Eulerian-Lagrangian algorithm can beexamined by solving (2) numerically, with the prescribed forc-ings s9(x, t), c90(x), and g9(x, t) set either to the Gaussian orsinusoidal Gaussian expressions given in (7) or (9). We canthen compare the resulting numerical solutions cG(x, t) orcSG(x, t) with the corresponding closed-form test solutionsc*G(x, t) or c*SG(x, t).

In order to show how the prescribed forcing method revealsnumerical errors, we have computed numerical approxima-tions to each of our two test solutions on a set of computationalgrids of increasing resolution. The coarsest grid (with a gridspacing of 2 m and 241 by 121 nodes) is selected as the refer-ence. The finer-resolution grids have grid spacings of 1.0, 0.5,and 0.25 m, respectively. The number of nodes for each ofthese cases is increased as required to keep the domain sizeconstant at 480 m by 240 m.

Concentration contour plots provide a convenient way toexamine the performance of the numerical algorithm at anygiven time. Plate 1 shows the concentration contours at thefinal simulation time (2400 days) obtained from the numericalsolutions to the (1) original problem (first column); (2) theproblem which should reproduce the Gaussian test solution(second column); and (3) the problem which should reproducethe sinusoidal Gaussian test solution (third column). Each rowcorresponds to one of the grid resolutions, with the coarsestgrid in the first row (from the top) and the finest grid in thefourth row. The exact (closed form) Gaussian and sinusoidaltest solutions are provided for reference in the fifth row.

It is revealing to note the changes in the plume shape andpeak concentration for the original problem (first column) asthe grid is refined. The fine-grid solver provides a complexplume with quite a bit of fine structure but no oscillations(oscillatory solvers typically generate negative concentrationson the fringes of the plume). The coarsest-grid solution ismuch smoother, with lower peak values, than the fine-gridsolution, suggesting the presence of numerical dispersion. Thisis typical of coarse-grid Eulerian-Lagrangian solvers which uselinear interpolators [Ruan and McLaughlin, 1999].

The effect of grid resolution on the accuracy of the numer-ical solution is confirmed by plots of the numerical approxi-mations to the two test solutions (second and third columns ofPlate 1). The approximate solutions differ significantly fromthe exact test solutions for coarse resolutions but converge tothe exact solutions as the grid resolution increases. The sinu-soidal Gaussian test case seems to be more sensitive to gridspacing since it converges more slowly. Even at the finestresolution, there are visible differences between the numericaland exact solutions. This is to be expected since the exactsinusoidal Gaussian test solution is more spatially variable andmore difficult to reproduce numerically than the exact Gauss-ian test solution. Of course, the sinusoidal Gaussian solutioncan be made still more difficult if the fluctuation length isdecreased or the fluctuation amplitude is increased.

Plate 1 nicely illustrates the practical advantage of using theprescribed forcing method as a substitute for a conventionalvariable-grid convergence analysis. The conventional conver-gence analysis reveals the presence of numerical dispersion at

coarser resolutions only when solutions are computed overgrids that are significantly finer than the reference (coarsest)grid. By contrast, the prescribed forcing method reveals nu-merical problems with only one simulation at the coarsestresolution.

Additional insight can be obtained from Figure 1, whichshows the log root-mean-square error measure log10 J(t) atseveral different times plotted versus log grid spacing log10 Dx .The prescribed forcing error is quite stable over time, indicat-ing that differences between the numerical and exact test so-lutions persist as the plume moves through the simulationdomain. It is clear from Plate 1 that the log error measuredecreases nearly linearly with decreasing log10 (Dx). Note thatthe error measure is significantly larger for the more difficultsinusoidal Gaussian case.

4. DiscussionThe prescribed forcing method described in this note is

clearly able to reveal inaccuracies in numerical solvers. It iseasy to implement, and it has the distinct advantage of beingable to test performance when coefficients, boundary condi-tions, and forcing terms vary over space and comparisons withanalytical solutions are not feasible. In its present form themethod is best suited to identifying the presence rather thanthe source of numerical errors. However, it is often possible todiagnose problems revealed by the method in an indirect fash-ion.

For example, the linear Eulerian-Lagrangian approach usedin our test problem is known to be numerically dispersive.Consequently, when the prescribed forcing method indicatesthat the numerical and exact test solutions differ, it is reason-able to suspect that numerical dispersion is the culprit. Anappropriate corrective measure in this case is to reduce thegrid spacing. The prescribed forcing method indicates that thisdoes, in fact, improve solution accuracy. Similarly, if the pre-scribed forcing method gives poor results when applied to anumerical algorithm that is prone to oscillation (such as a

Figure 1. The log10 of the root-mean-square error ( J) be-tween exact and computed test solutions for the Gaussian andsinusoidal Gaussian test problems, plotted at different timesversus log10 of the grid spacing (Dx).

3965RUAN ET AL.: TECHNICAL NOTE

quadratic Eulerian-Lagrangian approach), we can suspect thatoscillation is a problem. It would be useful if we could makethis diagnostic process more precise, perhaps by a more de-tailed analysis of the errors between the exact and approximatetest solutions. However, this is difficult to do with any gener-ality. Part of the problem is that the spatial pattern of theerrors is affected by the structure of the heterogeneous velocityfield. This confounding influence needs to be taken into ac-count in a detailed spatial error analysis.

It is reasonable to ask if we could extract more informationfrom the prescribed forcing method if we looked at a widerrange of test solutions. Certainly, it is possible that more can bedone. In fact, this would be a useful topic for further research.However, we feel that the information provided by the presentversion of the method is sufficient to assess performance inpractical situations where model inputs are highly variable andthe true solution is unknown. If the method reveals problems,changes can be made, and the prescribed forcing test can berepeated until the problems are resolved. In the case of solutetransport we feel that the prescribed forcing method providesa more realistic performance test than other criteria (such as acheck of the grid Peclet number) which are based on constantvelocity assumptions or other simplifications.

In closing, it is worth noting that the prescribed forcingmethod is quite general. It is not confined to closed-form testsolutions or even to test solutions that satisfy particular differ-ential equations (such as the transport equation). The advan-tage of using a closed-form solution is that numerical errorsare contributed only by the algorithm used to solve the differ-ential equation of interest (criterion 1). The advantage of usinga test solution obtained from a differential equation similar instructure to the original equation is that it is generally easier toinsure that the prescribed forcing terms do not dominate thenumerical problem (criterion 2). However, these are not strictrequirements, and we may choose to use any reasonable testsolution, so long as the prescribed forcing terms can be accu-rately computed and are small enough to insure that the testingprocedure checks the algorithm’s ability to solve the originalproblem. We encourage modelers and numerical analysts totry the prescribed forcing method and to document their ex-periences so that the capabilities and limitations of the methodcan be better defined.

Acknowledgment. The authors acknowledge the helpful insightsand comments provided by Carsten Schwarz and Wolfgang Kinzel-bach.

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(Received October 13, 1998; revised July 6, 1999;accepted July 30, 1999.)

RUAN ET AL.: TECHNICAL NOTE3966