Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol23.Issue1.1999:

An evaluation of temporally adaptive

transformation approaches for solving Richards'

equation

Glenn A. Williams

*

_{& Cass T. Miller}

Department of Environmental Sciences and Engineering, University of North Carolina, Chapel Hill, NC 27599-7400, USA

(Received 15 May 1998; revised 28 October 1998; accepted 15 November 1998)

Developing robust and ecient numerical solution methods for Richards' equa-tion (RE) continues to be a challenge for certain problems. We consider such a problem here: in®ltration into unsaturated porous media initially at static con-ditions for uniform and non-uniform pore size media. For ponded boundary conditions, a sharp in®ltration front results, which propagates through the media. We evaluate the resultant solution method for robustness and eciency using combinations of variable transformation and adaptive time-stepping methods. Transformation methods introduce a change of variable that results in a smoother solution, which is more amenable to ecient numerical solution. We use adaptive time-stepping methods to adjust the time-step size, and in some cases the order of the solution method, to meet a constraint on nonlinear solution convergence properties or a solution error criterion. Results for three test problems showed that adaptive time-stepping methods provided robust solutions; in most cases transforming the dependent variable led to more ecient solutions than un-transformed approaches, especially as the pore-size uniformity increased; and the higher-order adaptive time integration method was robust and the most ecient method evaluated. Ó _{1999 Elsevier Science Ltd. All rights reserved}

1 INTRODUCTION

Modeling of variably saturated ¯ow is an important problem of practical interest for which signi®cant issues remain unresolved. Among them are the appropriate formulation of governing equations and constitutive relations1,2. While important formulation issues remain, the standard approach to model variably saturated ¯ow is through the use of numerical solution to Richards' equation (RE). The amount of work done on numerical solutions to RE notwithstanding, signi®cant issues of robustness and eciency remain for certain classes of dicult test problems, especially those that give rise to sharp fronts that propagate through the domain. The numerical simulation of this type of problem can be computationally challenging due to the ®ne spatial and

temporal discretizations required to resolve the sharp front accurately.

The most common approaches for approximating RE use low-order ®nite di€erence or ®nite element spatial approximations and low-order time integration3±17. In addition, most variably saturated ¯ow simulators cur-rently in use are based upon ®xed spatial grids and either ®xed step or an empirically based adaptive time-stepping (EBATS) method14,13. Because EBATS procedures are not based on estimation of temporal truncation error, they are not able to control that error speci®cally in the solution.

A di€erential algebraic equation-based method of lines (DAE/MOL) solution of RE results in more robust and ecient solution to RE than traditional approach-es18. In this approach, estimates of temporal truncation error were used explicitly to control the solution order, which ranged from ®rst to ®fth order in time, and time-step size.

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*

Corresponding author. E-mail: glenn_williams@unc.edu

Transformation methods for RE can also lead to a more robust and ecient numerical approximation than traditional approaches19; the inherent nonlinearity is reduced by applying a change of variables to the de-pendent variable. The solution of the original problem may then be retrieved via an inverse transformation. This approach has been analysed and tested for a wide variety of test problems using ®xed time-step integra-tion19. Because transformation of the dependent vari-able leads to a smoother front in space and time, we conjecture that combining temporally adaptive, higher-order time integration methods applied to a transformed form of RE will lead to a robust and ecient solution strategy for RE. This combination of approaches has not been examined in the literature to our knowledge.

The objectives of this work were (1) to identify combinations of transformation and temporally adap-tive methods that yield robust and ecient solutions to RE; (2) to compare the eciency of the adaptive time-stepping transformation (ATST) methods with current state-of-the-art approaches for a range of test problems; (3) to evaluate the sensitivity of the solution robustness and eciency to the value of the transform parameter; and (4) to examine the potential for de®ning an esti-mator for the transform parameter that will provide near-optimal performance for ATST methods.

2 BACKGROUND

The numerical solution of RE requires decisions about the form of the equation to be solved, the constitutive relations used to close the equation, the spatial ap-proximation, the temporal apap-proximation, the nonlinear equation solution, and the linear equation solution methods. Standard approaches have evolved for each of these decisions, although recent advancements o€er potentially attractive alternatives to the standard choices in some cases.

Several forms of RE are possible: the pressure-head, moisture-based, mixed, or other transformed forms of the equation20,12,14,21. Choosing the appropriate form of the governing equation is related to the method of re-solving the nonlinearities present in the accumulation term, since mass conservation problems may result un-less care is taken22,23,14. Popular choices include the mixed-form equation ± using either modi®ed Picard it-eration (MPI)14 or Newton iteration (NI)11,15,16 to re-solve the nonlinearities ± or chord iteration to approximate the speci®c moisture capacity term in the pressure-head form along with Picard iteration (PI) to resolve the nonlinearities24. Recent work has shown that mass conservation problems may be overcome even if the pressure-head form of RE is used, if formal error control is used to manage temporal integration18.

Transformation methods have long been used to aid in the solution of RE20,25±27,21,16,28. The goal of

trans-formation methods is to make the numerical solution more ecient by reducing the strong nonlinearity of the media hydraulic properties as functions of pressure. Various transformations have been successfully applied to the solution of RE, including integral20, water-con-tent-based21,16, rational function28, and hyperbolic sine transforms26. A comprehensive investigation of trans-formation methods was recently completed and an ef-fective new transform (IT2) introduced19. IT2 is a combination of integral and water-content-based transforms and was found ecient and robust for a range of test problems compared to all other existing transforms for a wide range of media properties and discretization levels19. However, this investigation fo-cused on ®xed time-step methods.

The use of low-order ®nite di€erence29,20,12,14,24,17or ®nite element10,30,11,23,13±15,24,16methods to approximate the spatial derivatives in RE are the dominant ap-proaches. These methods are usually applied on a ®xed spatial grid, although an adaptive approach has been examined as well13. Because sharp fronts in space and time can exist for certain problems solved with RE, adaptive approaches in space and time are appealing for this class of problem.

The standard temporal approximation method used to approximate RE is the one-step Euler approach14. The most common solution approaches use a fully im-plicit (backward-di€erence) time approximation14,24,31,19 which has a truncation error of O…Dt†. Accurate, or under some circumstances even convergent, solutions to RE often require very small time steps over a portion of the simulation when using standard approaches. An adaptive time-stepping scheme is often implemented in variably saturated ¯ow simulators to optimize time-step requirements32±35. These adaptive time-stepping meth-ods are usually based on empirical approaches to de-termine when and by how much to adjust the time-step size.

Recent work has shown that variable step size, vari-able order time integration methods can lead to robust and ecient codes using a DAE/MOL approach18,31. In this approach, the partial di€erential equation, RE, is reduced to a system of ordinary di€erential equations (ODEs) in time by approximating the spatial derivatives using standard approaches (e.g., ®nite di€erence or ®nite elements methods) and then integrated in time using a DAE code. One advantage of the DAE/MOL approach is that the error checking and control, and adaptive time-stepping are handled by a mature and available DAE solver, although some modi®cations to standard approaches have been found useful18.

from application of NI are often poorly conditioned18_, which can lead to nonlinear convergence failures or loss of accuracy in the solution itself. A backtracking line search procedure38can be implemented to improve the global convergence properties of NI methods. A qua-dratic/cubic line search has been used successfully in concert with NI18,19, which we term an NI-line search (NILS) approach. NILS is much more robust than NI18. The implicit solution of RE requires the solution of systems of linear equations, which is typically done using direct elimination methods in one dimension10,11, although cyclic reduction is much more ecient on vector machines18. In multiple dimensions for the stan-dard spatial and temporal approximation methods, sparse matrices result, but bandwidths can become large. In this case, iterative approaches, such as Bi-CGSTAB or GMRES, must be used to solve large-scale problems39.

3 APPROACH 3.1 Formulation

Because the main considerations of this work were not dependent upon the spatial dimensionality of the ap-proach, we used a one-dimensional approach for sim-plicity. RE may be formulated in several ways40,22,41,14. To facilitate our use of transformation methods, we introduce two general statements of RE, the p-based form

and the mixedp-based form

o_ha

where c is a speci®c moisture capacity, Ss the speci®c storage coecient, which accounts for ¯uid compress-ibility,Sa saturation of the aqueous phase; p a general transformation variable such that pˆp…w†, w the pressure head,ttime,hathe volumetric water fraction of the aqueous phase,zthe vertical spatial dimension, and K the hydraulic conductivity.

We consider problems with auxiliary conditions of the form

p…z;tˆ0† ˆp0…z† ˆp…w0…z††; …3† p…zˆ0;t>0† ˆp1ˆp…w1†; …4† p…zˆZ;t>0† ˆp2ˆp…w2†; …5† where Z is the length of the domain, w0 may be a function of space, and w1 and w2 are constants. The auxiliary conditionsp0,p1, and p2 are found by the ap-propriate change of variables of the initial and boundary conditions given in terms ofw. We consider these aux-iliary conditions because they can lead to a sharp

in®l-tration front and saturated conditions over a portion of the domain, which is a dicult class of test problem.

3.2 Constitutive relations

Constitutive relations must be speci®ed to close RE. We used the standard van Genuchten (VG) pressure±satu-ration relationship42, which is given by

Se…w† ˆha…w† ÿhr irreducible volumetric water content, hs the saturated volumetric water content,ava parameter related to the mean pore size, and nv a parameter related to the uni-formity of the pore-size distribution.

The speci®c moisture capacity,c, is de®ned as dhadw. Using eqn (6) we see that,

c…w† ˆ

We used Mualem's model for the relative permeability of the aqueous phase43

K…Se† ˆKsSe1=2‰1ÿ …1ÿS 1=mv

e †

mv

Š2; …8†

where Ks is the water-saturated permeability, and Seˆ

Se…w†from eqn (6).

3.3 Transformations

The objective of transformation methods is to de®ne a function p…w† that will result in a more ecient and robust solution to the governing equation, eqn (1). This is accomplished by introducing a change of dependent variable that results in the solution in terms ofp being smoother and less sharp than a solution in terms of w. The original problem's solution is then retrieved by applying an inverse transformation. Results show that IT2 transform performs well over a wide range of media and auxiliary conditions19. A rational function trans-form (RFT)28also performs well for many problems and was used for comparison purposes in this work.

IT2 is de®ned by

pˆ

wherebis a free parameter. RFT is de®ned by

pˆ

3.4 Spatial discretization

We use a standard ®nite-di€erence approximation14 to discretize RE with respect to the spatial dimension, z, where z2 ‰0;ZŠ. We consider a uniform spatial discret-ization comprised of nnÿ1 intervals of length Dz, with

DzˆZ…nnÿ1†, and ziˆ …iÿ1†Dz for 16i6nn. The

wherenn is the number of spatial nodes in the solution, andpi is the approximation top…zi†.

3.5 Empirical time adaption

The most common approaches for adaptive time-step solutions to RE use an empirically based (EBATS) scheme32±35. The EBATS approach we used was imple-mented according to the algorithm

ÿif ml6m6mu then Dtn‡1 Dtn‡1

ÿelse if m<ml thenDtn‡1ˆmin…ftDtn;Dtmax†

ÿelse; sincem>mu; then Dtn‡1

ˆmax…Dtn ft;Dtmin†

where m is the number of iterations required by the nonlinear solver to converge for time stepn,mla lower iteration limit,muan upper iteration limit,fta time-step acceleration factor,Dtmaxthe maximum allowable time-step size, and Dtmin the minimum allowable time-step size.

Although this empirical approach can be easily im-plemented in existing ®xed time-step codes, it requires the speci®cation ofml,mu,ft,Dtmin, andDtmaxfor which theoretical guidance does not exist. In this work, Dtmin, and Dtmax were chosen to yield a set of convergent so-lutions with a wide range of accuracies. The remaining parameters were set by trial and error to yield robust and ecient solutions.

3.6 DAE/MOL time integration

We investigated a higher-order DAE/MOL approach applied to the p-based form of RE eqn (1)18_{, which in}

This set of equations may be solved by an implicit ODE or DAE integrator, with a sti€ solver being the most reasonable choice. In this work, we use the package DASPK44, a DAE integrator based on a ®xed leading coecient BDF method of variable step size and order up to ®fth. This code is an extension of the popular DAE integrator DASSL, which is available through netlib (http://www.netlib.org/). The error checking, or-der selection, and time-step adaptivity features available in DASPK can be applied to the time integration of RE to achieve a pre-speci®ed level of temporal accuracy. In recent work, we detailed this solution approach and compared eciency with a variety of standard ap-proaches18,31. Brenanet al.,45 present a detailed discus-sion of the BDF order and step-size selection used in DASPK.

3.7 Interblock permeability estimation

An important aspect of the numerical solution process is the approach used to estimate permeabilities within the spatial discretization scheme. The values of concern appear as Ki1=2 in eqn (12). Several approaches have been suggested and compared in the literature46±48, but an integral approach for evaluating Ki1=2 was recently shown to be ecient and robust31. This is particularly important in cases involving non-uniform pore size media, where the permeability function as de®ned by the van Genuchten/Mualem (VGM) model becomes non-smooth. For simulation of variably saturated ¯ow in non-uniform media, non-integral interblock permeabil-ity estimation methods such as arithmetic averaging of nodal values often leads to convergence failure of the nonlinear solver31. Therefore, we will use the integral technique in this work in order to ensure ecient and robust solutions.

The integral estimation method (KINT) for the p version of RE eqn (1) is expressed19as

Ki1=2

This approach is not routinely used to solve the un-transformed RE due to the apparent computational expense. However, an integral approach can be used to solve the transformed RE eciently by taking advan-tage of tabulation and accurate interpolation methods19.

3.8 Error tolerance

dependent variable, which isp in the transformed case. For highly nonlinear transforms such as IT2,p values can be extremely small (10ÿ18₁₀ÿ6_{) over a majority of} the w domain. This will have a direct e€ect on error tolerances and termination of the nonlinear solver.

In DASPK, the speci®ed relative and absolute error tolerances (rtol and atol, respectively) are used by the code in a local error test at each step which requires that

jlocal error inpij6EWT…i†; …16†

where

EWT…i† ˆrtol jp…i†j ‡atol …17†

is a combined, allowable absolute and relative error. For transforms such as IT2, the local error test will not be stringent enough in portions of the domain where piis very small unless atol is a set to a very small value.

We experimented with various atol values and achieved the most accurate results by setting atolˆ0, which represents a strictly relative error tolerance. By specify-ing a range of values for rtol, a range of accuracies in the resultant solution and computational e€ort needed for the solution were obtained.

3.9 Eciency considerations and evaluation

The analytic evaluation of constitutive relations repre-sents a signi®cant portion of the computational e€ort required to solve RE numerically, due to the compli-cated power functions involved. To increase the e-ciency of the overall simulation, these relations are often evaluated by tabulating a set of analytic values and then interpolating intermediate values as they are required during the simulation. Linear interpolation is often used, yet higher-order methods such as cubic or Hermite spline interpolation may be required for higher-accuracy solutions31. This tabulation and interpolation procedure results in signi®cant savings in computational e€ort without lost accuracy, compared to direct function evaluations. Based upon our previous work, which provides a detailed description of the approach31,39, we used a Hermite spline interpolation procedure to ensure robust and accurate solutions.

Error vs. work plots illustrate the eciency of the solution methods. In order to compare various solution methods, some measure of work is required. To aid generality of our results beyond the platform in which the experiments were performed, we de®ne work in the manner described below. As a basis of comparison however, typical CPU times were on the order of a few minutes per simulation on an HP 9000/715 workstation. For the DASPK solver, which relies upon Newton iteration to resolve nonlinearities, the majority of the work is associated with Jacobian evaluations, function evaluations, and the solution of the linear system of equations. This observation allows for a simple, straightforward measure of work that requires relative

weights for the three procedures and integer counts for each of the procedures, such as

Wnˆwjnj‡wfnf‡wlnl; …18†

whereWnis a work measure for Newton iteration DAE methods, wj a weighting factor for formation of the Jacobian matrix,wf a weighting factor for evaluation of the function, wl a weighting factor for solution of the linear system of equations, nj the number of Jacobian evaluations,nf the number of function evaluations, and nl the number of the linear solutions performed. From previous work, we obtained estimates of these weighting coecients for KINT/DASPK solutions to the un-transformed RE using ®nite-di€erence approximation to evaluate the Jacobians and Hermite spline interpolation of constitutive relations31:…wj†utˆ0:631,…wf†utˆ0:311, and …wl†utˆ0:181. From subsequent work on the transformed RE, we obtained the following estimates for the same solution methods listed above but applied to the transformed instead of the untransformed RE:

…wj†trˆ0:883, …wf†trˆ0:435, and …wl†trˆ0:181. These weighting factors result in approximately 30% addi-tional work for the transformed solution compared to the untransformed solution.

Error was evaluated by comparison to a dense-grid solution. This error, referred to as dense-grid …D†, is de®ned by

jjDjjk ˆ

1 nn

Xnn

iˆ1

…jw^_i "

ÿw_ij†k

#1=k

; …19†

wherekis the norm measure, andw^i is an accurate

ap-proximation of the true solution based on a dense spa-tial grid.kˆ1,kˆ2, andkˆ 1were considered in this work and termed L1, L2, and L1 error norms, respec-tively. The dense-grid solutions were generated using the MPI solver with temporal and spatial grid sizes equal to 1/32 of the standard sizes used in the test simulations.

3.10 Parameter optimization

Optimal values for the parameter,b, in the IT2 or RFT transforms can be found by optimizing some perfor-mance-based objective function such as amount of work required or dense-grid error. In this work, we de®ne the objective function as the product of work and dense-grid error. Thus, for a given simulation, the optimalbvalue

…bopt†is found by solving the constrained minimization problem

min

bmin6b6bmax

fWn…b† jjD…b†jj1g; …20†

whereb is the arbitrary transform parameter, bmin and bmax are the minimum and maximumbvalues resulting in convergence of the nonlinear solver,Wnis the required work as de®ned by eqn (18), and jjDjj1 is the L1 error norm as de®ned by eqn (19).

projected quasi-Newton algorithm that uses a decreas-ing sequence of ®nite di€erence steps (scales) to ap-proximate the gradient. It uses an approximation to the Hessian and a line search algorithm that gives the code global convergence capabilities under certain conditions. This algorithm reduces the chance compared to a con-ventional Newton method that an optimal result will be returned at a local minimum rather than a global min-imum.

4 RESULTS AND DISCUSSION 4.1 Test problems

We compared the two temporally adaptive approaches for solving RE for both the transformed and untrans-formed case for three test problems. The simulation conditions for the three test problems (A, B, and C) are given in Table 1, including constitutive relationship parameters, spatial and temporal domains, auxiliary conditions, and standard spatial and temporal discreti-zation intervals. These test problems represent a variety of media and auxiliary conditions. The material prop-erties for Problem A correspond to a dune sand50. Problem B represents a clay material and has served as a test problem in previous work14,24,18. The material properties in Problem C correspond to the average values for the soil textural group loam according to the USDA classi®cation51 as estimated by Carsel and Par-rish52from analyses of a large number of soils.

Problems A and C are dicult to simulate numeri-cally due to the relatively dry initial conditions and the steep wetting fronts that develop. These problems pro-vided a rigorous test for the methods outlined in this work. Problem B is substantially easier than A or C because the domain is much smaller, the media is

par-tially saturated inipar-tially, and fully saturated conditions do not develop. Yet, Problem B allowed us to compare our results with recent research performed using state-of-the-art methods. Example coarse-grid and dense-grid solution pro®les for all problems are shown in Fig. 1. For each problem, signi®cant errors can be observed in the vicinity of the wetting front. Note that the front is somewhat sharper for Problems A and C than for Problem B as a result of the boundary conditions.

4.2 Performance comparisons

A set of simulations was conducted to compare perfor-mance for four di€erent solution approaches: an EBATS solution (as outlined in Section 3.5) of the un-transformed RE; a DASPK solution of the untrans-formed RE; an EBATS solution of the transuntrans-formed RE; and a DASPK solution of the transformed RE. The EBATS results were generated using an MPI solver applied to the standard mixed form of RE. The DASPK solutions were generated using up to ®fth order BDF methods in time applied to the pressure-head form of RE. Interblock permeabilities using the KINT approach were evaluated using four-point Gauss±Legendre quadrature. A range of solution accuracies were pro-duced by varying the temporal discretization parameters for the EBATS model and the rtol values in the DASPK solution.

Figs. 2±4 show the results of these simulations in the form of error vs. work plots for Problems A±C, re-spectively; results shown for the transformed RE were generated using the IT2 transform. U and T in Figs. 2±4 designate untransformed and transformed solutions, respectively. Non-monotonic trends in solution error as a function of work resulted for some of these simula-tions; this trend was especially true for the DASPK solver. This trend is a result of the heuristic methods used to change solution order and step size and because

Table 1. Test problem simulation conditions

Variable Problem A Problem B Problem C

hr (±) 0.093 0.102 0.078

hs (±) 0.301 0.368 0.430

av…mÿ1_{) 5.47 3.35 3.60}

nv (±) 4.264 2.000 1.560

Ks (m/day) 5.040 7.970 0.250

Ss (mÿ1) 1:00106 0.00 1:0010ÿ6

z(m) [0,10.0] [0,0.3] [0,5.0]

t(days) [0,0.18] [0,0.092] [0,2.25]

w₀ (m) ÿz ÿ10.00 ÿz

w₁ (m) 0.00 ÿ10.00 0.00

w₂ (m) 0.10 ÿ0.75 0.10

Dz (m) 0.0125 0.0025 0.025

Dtmin (days) 110ÿ10 110ÿ10 110ÿ5

Dtmax(days) 110ÿ2 110ÿ2 210ÿ1

ft 1.2 1.2 1.2

ml [1,15] [1,15] [1,15]

mu [6,50] [5,30] [10,200]

Fig. 1.Comparison of solution pro®les for dense-grid (lines)

error is controlled in the DASPK solution over a time-step and only by inference to solver e€ort in the EBATS approaches.

Based upon this set of simulations, we made the following observations:

(i) the DAE/MOL approach using DASPK was more e€ective than the empirically based time-step ap-proach for intermediate to high levels of accuracy; (ii) transformed DASPK solutions tended to be in-creasingly ecient compared to untransformed DASPK solutions asnvincreased, which corresponds to increasingly sharp in®ltration fronts for identical auxiliary conditions;

(iii) transformed solutions achieved lower errors than untransformed solutions for both time integration methods for Problem A, which we attribute to the sharp nature of the front in untransformed space; (iv) the RFT transform was not as robust or ecient as the IT2 transform for the test problems considered; and

(v) no loss of mass balance was seen for the higher-or-der DASPK/transform solutions as compared to that of the higher-order DASPK/untransformed solutions. These data show that for solving RE, the IT2 transform combined with the more sophisticated error checking, order selection, and step-size selection in DASPK o€ers signi®cant computational advantages. Optimal perfor-mance of this solution approach will depend on the IT2 parameterb.

4.3 Parameter sensitivity

To evaluate the sensitivity of the transform parameter, bopt values for IT2 were found using IFFCO for a wide range of media and discretization parameters. For each of the three test problems, we conducted parameter optimization experiments for DASPK solutions using IT2 to identify bopt values as the parameters

Dz; av; andnv were varied. In addition, an optimal range was calculated for each case, where the lower and upper end of the range represent b values where the resulting objective function was within 10% of that at bopt. The lower and upper limits of this range are de®ned asbÿ₉₀ andb‡₉₀, respectively. Tables 2±4 show the results of these parameter optimization experiments. Due to varying lengths of spatial domains, only selected spatial discretizations were analyzed for each problem, which results in some blanks in Table 2.

Analysis of the parameter optimization data resulted in the following observations:

(i) for the easier test problem, B, where saturated con-ditions did not develop,boptwas relatively insensitive to changes inDz; av; andnv;

(ii) in general,boptwas more sensitive toDzin regions of relatively ®ne or coarse spatial discretizations and less sensitive in regions of intermediate spatial discret-ization scales.

(iii) for non-uniform media …nv62:0†; bopt ap-proached zero and was relatively insensitive to chan-ges in media parametersav andnv;

Fig. 4.Error vs. work comparisons for Problem C.

Fig. 2.Error vs. work comparisons for Problem A.

(iv) for dicult, initially dry, sharp-front problems such as A,boptappeared to be more sensitive to vari-ations innvthan to variations in av; and

(v) the performance of the DASPK solver, as mea-sured by the objective function workerror, was less sensitive to small deviations form bopt than the per-formance of ®xed time-step solvers.

4.4 Parameter estimation

The parameter sensitivity data did not produce a well-de®ned estimate of the optimal transform parameter that was reliable over the entire range of media and spatial discretization parameters. It did, however, pro-vide some insight for guiding selection of transform

parameters for several classes of problems. For prob-lems involving lownvvalues…62:0†or where saturated conditions do not develop, smallb values on the order of 0.001±0.01 appear to work well over a wide range of media and discretization parameters. For problems where saturated fronts develop in initially dry media that is relatively uniform…nv>2:0†, de®ning a reliable estimator for the transform parameter is more compli-cated. Based on our numerical experiments, it appears that an estimator for these types of problems must in-volve some function ofDzandnv. We did observe that optimal b values for these types of problems generally fall in the range of 0.01±0.3, and performance of the DASPK solver did not degrade signi®cantly for deviations form bopt within this range. Therefore, if

Table 3. Optimal Parameter Range vs.a_v

av Problem A Problem B Problem C

bÿ₉₀ bopt b‡90 b

ÿ

90 bopt b‡90 b

ÿ

90 bopt b‡90

2.0 0.030 0.060 0.075 0.001 0.001 0.002 0.001 0.001 0.002

5.0 0.040 0.050 0.085 0.003 0.004 0.008 0.001 0.001 0.002

8.0 0.015 0.020 0.030 0.001 0.003 0.008 0.008 0.008 0.010

1.0 0.017 0.020 0.033 0.001 0.001 0.006 0.002 0.004 0.005

3.0 0.046 0.050 0.080 0.001 0.002 0.007 0.002 0.002 0.005

Table 2. Optimal Parameter Range vs. Spatial Discretization

Dz…10ÿ2_{† Problem A Problem B Problem C}

bÿ₉₀ bopt b‡90 b

ÿ

90 bopt b‡90 b

ÿ

90 bopt b‡90

0.98 0.080 0.100 0.450 0.030 0.065 0.100

1.25 0.050 0.080 0.140 0.001 0.003 0.006 0.036 0.044 0.060

1.56 0.050 0.070 0.100 0.007 0.010 0.015

1.95 0.040 0.060 0.120 0.003 0.006 0.007

2.50 0.040 0.050 0.100 0.003 0.006 0.008 0.001 0.004 0.006

3.13 0.030 0.060 0.110 0.001 0.002 0.005 0.001 0.002 0.003

3.75 0.002 0.003 0.005

4.69 0.001 0.001 0.007

5.00 0.050 0.060 0.100 0.001 0.002 0.006 0.001 0.001 0.002

6.25 0.030 0.050 0.090 0.002 0.003 0.006 0.010 0.013 0.017

7.81 0.040 0.050 0.090 0.004 0.005 0.008

9.38 0.001 0.002 0.006

10.00 0.005 0.009 0.010 0.001 0.002 0.005 0.002 0.002 0.005

12.50 0.001 0.002 0.008 0.001 0.001 0.003

20.00 0.001 0.001 0.006 0.001 0.001 0.003

Table 4. Optimal Parameter Range vs.n_v

nv Problem A Problem B Problem C

bÿ₉₀ b_opt bÿ₉₀ bÿ₉₀ b_opt b‡₉₀ bÿ₉₀ b_opt b‡₉₀

1.5 0.001 0.001 0.002 0.001 0.001 0.002 0.001 0.001 0.002

2.0 0.004 0.006 0.007 0.003 0.006 0.010 0.001 0.001 0.002

2.6 0.080 0.110 0.140 0.001 0.001 0.007 0.001 0.002 0.003

3.2 0.170 0.180 0.185 0.001 0.006 0.010 0.002 0.003 0.005

3.8 0.210 0.310 0.390 0.001 0.001 0.006 0.001 0.001 0.002

near-optimalperformance is acceptable,b values within the 0.01±0.3 range will usually produce acceptable re-sults for these types of problems. This near-optimal performance is usually more ecient than that of al-ternative solution strategies, such as untransformed, ®xed step, or empirically based adaptive time-stepping.

5 CONCLUSIONS

We can draw several conclusions based on our work. · _{We outline a set of computational techniques ±}

in-cluding an MOL formulation, a recently intro-duced transform (IT2), an integral approach for estimation of interblock permeabilities, and an e-cient Hermite spline interpolation procedure for function evaluation ± that permits ecient and ro-bust application of the DAE solver DASPK to dif-®cult sharp-front in®ltration problems.

· _{Transformation methods combined with DAE/}

MOL approaches are generally more ecient than transformation methods applied to ®xed or empir-ically adaptive time-step approaches.

· _{The steep nature of the error±work relationship for}

the DASPK/IT2 solver used in this work increases the relative eciency of this method compared to low-order temporal integration methods for inter-mediate to high levels of accuracy.

· _{The potential bene®t of a particular transform}

de-pends upon speci®cation of an appropriate value for the transformation parameter, and results show that when using the IT2 transform with the DAE solver DASPK, this selection may be simpli®ed due to the relative insensitivity of solver perfor-mance to deviations from the optimal parameter.

ACKNOWLEDGEMENTS

This work supported in part by US Army Waterways Experiment Station Contract DACA39-95-K-0098, Ar-my Research Oce Grant DAAL03-92-G-0111, Na-tional Institute of Environmental Health Sciences Grant 5 P42 ES05948, and a Department of Energy Compu-tational Science Fellowship. Computing activity was partially supported by allocations from the North Car-olina Supercomputing Center.

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