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Advances in Water Resources Vol. 22, No. 8, pp 831±840, 1999
Ó 1999 Elsevier Science Ltd
Printed in Great Britain. All rights reserved
0309-1708/99/$ ± see front matter

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' equation (RE) continues to be a challenge for certain problems. We consider such a
problem here: in®ltration into unsaturated porous media initially at static conditions 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 untransformed 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

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 currently in use are based upon ®xed spatial grids and either
®xed time-step or an empirically based adaptive timestepping (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 approaches18. 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 timestep size.

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

*

Corresponding author. E-mail: glenn_williams@unc.edu
831

832

G. A. Williams, C. T. Miller

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 dependent 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 integration19. Because transformation of the dependent variable leads to a smoother front in space and time, we
conjecture that combining temporally adaptive, higherorder 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 adaptive methods that yield robust and ecient solutions to
RE; (2) to compare the eciency of the adaptive timestepping 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 estimator 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 approximation, the temporal approximation, 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 resolving the nonlinearities present in the accumulation
term, since mass conservation problems may result unless care is taken22,23,14. Popular choices include the
mixed-form equation ± using either modi®ed Picard iteration (MPI)14 or Newton iteration (NI)11,15,16 to resolve 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-content-based21,16, rational function28, and hyperbolic sine
transforms26. A comprehensive investigation of transformation methods was recently completed and an effective 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 focused on ®xed time-step methods.
The use of low-order ®nite di€erence29,20,12,14,24,17 or
®nite element10,30,11,23,13±15,24,16 methods to approximate
the spatial derivatives in RE are the dominant approaches. 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 implicit (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 methods are usually based on empirical approaches to determine when and by how much to adjust the time-step
size.
Recent work has shown that variable step size, variable 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.
PI12,14, MPI14,21, and NI11,15 are far and away the
most frequently used methods to resolve the nonlinearities in RE. While the former two converge linearly, NI
converges quadratically in the vicinity of the solution36,37, which can lead to more ecient solutions when
high accuracy is desired. Jacobian matrices resulting

An evaluation of temporally adaptive transformation approaches for solving Richards' equation
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 procedure38 can be implemented to improve the
global convergence properties of NI methods. A quadratic/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 standard spatial and temporal approximation methods,
sparse matrices result, but bandwidths can become
large. In this case, iterative approaches, such as BiCGSTAB or GMRES, must be used to solve large-scale
problems39.

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±saturation relationship42, which is given by

ha …w† ÿ hr …1 ‡ jav wjnv †ÿmv w < 0;
…6†
Se …w† ˆ
1
w P 0;
hs ÿ hr
where mv ˆ 1 ÿ 1 nv , Se is the e€ective saturation, hr the
irreducible volumetric water content, hs the saturated
volumetric water content, av a parameter related to the
mean pore size, and nv a parameter related to the uniformity of the pore-size distribution.
The speci®c moisture capacity, c, is de®ned as dha dw.
Using eqn (6) we see that,
c…w† ˆ
(
…hs ÿ hr †mv …1 ‡ jav wjnv †ÿmv ÿ1 nv av jav wjnv ÿ1

3 APPROACH

0

3.1 Formulation

833

w < 0;
w P 0:
…7†

Because the main considerations of this work were not
dependent upon the spatial dimensionality of the approach, we used a one-dimensional approach for simplicity. 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



ow op

o
ow op
‡1
…1†
ˆ
K…p†
‰c…p† ‡ Ss Sa …p†Š
op ot oz
op oz
and the mixed p-based form



oha
ow op
o
ow op
‡ Ss Sa …p†
‡1 ;
…2†
ˆ
K…p†
ot
op ot oz
op oz

where c is a speci®c moisture capacity, Ss the speci®c
storage coecient, which accounts for ¯uid compressibility, Sa saturation of the aqueous phase; p a general
transformation variable such that p ˆ p…w†, w the
pressure head, t time, ha the volumetric water fraction of
the aqueous phase, z the vertical spatial dimension, and
K the hydraulic conductivity.
We consider problems with auxiliary conditions of
the form
p…z; t ˆ 0† ˆ p0 …z† ˆ p…w0 …z††;

…3†

p…z ˆ 0; t > 0† ˆ p1 ˆ p…w1 †;
p…z ˆ Z; t > 0† ˆ p2 ˆ p…w2 †;

…4†
…5†

where Z is the length of the domain, w0 may be a
function of space, and w1 and w2 are constants. The
auxiliary conditions p0 , p1 , and p2 are found by the appropriate change of variables of the initial and boundary
conditions given in terms of w. We consider these auxiliary conditions because they can lead to a sharp in®l-

We used Mualem's model for the relative permeability
of the aqueous phase43
m 2

K…Se † ˆ Ks Se1=2 ‰1 ÿ …1 ÿ Se1=mv † v Š ;
…8†
where Ks is the water-saturated permeability, and Se ˆ
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 of p 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 transform (RFT)28 also performs well for many problems and
was used for comparison purposes in this work.
IT2 is de®ned by
8
w
>
< R K…w0 † dw0 ‡ b ‰h…w† ÿ h Š w 6 0;
r
…9†
p ˆ ÿ1
>
: op j

w
‡
p…0†
w
>
0;
ow wˆ0
where b is a free parameter.
RFT is de®ned by
 w
w < 0;
p ˆ 1‡bw
w
w P 0;

where b is a free parameter.

…10†

834

G. A. Williams, C. T. Miller
where

3.4 Spatial discretization
14

We use a standard ®nite-di€erence approximation to
discretize RE with respect to the spatial dimension, z,
where z 2 ‰0; ZŠ. We consider a uniform spatial discretization comprised of nn ÿ 1 intervals of length Dz, with
Dz ˆ Z…nn ÿ 1†, and zi ˆ …i ÿ 1† Dz for 1 6 i 6 nn . The
spatial operator



o
ow op
‡1
…11†
K…p†
Os …p† ˆ
oz
op oz
is approximated at z ˆ zi for 1 < i < nn by
Osi …p†
ˆ Dzÿ1

†
…p ÿ pi † ÿ …K ow
†
…p ÿ piÿ1 †
…K ow
op i‡1=2 i‡1
op iÿ1=2 i
Dz
!

‡ Ki‡1=2 ÿ Kiÿ1=2 ;

…12†

where nn is the number of spatial nodes in the solution,
and pi is the approximation to p…zi †.


ow
:
…14†
op i
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, order 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 approaches18,31. Brenan et al.,45 present a detailed discussion of the BDF order and step-size selection used in
DASPK.
A…p†i ˆ ‰c…p† ‡ Ss Sa …p†Ši



3.7 Interblock permeability estimation

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 implemented according to the algorithm
ÿ if ml 6 m 6 mu then Dtn‡1 Dtn‡1
ÿ else if m < ml then Dtn‡1 ˆ min…ft Dtn ; Dtmax †
ÿ else; since m > mu ; then Dtn‡1
ˆ max…Dtn ft ; Dtmin †
where m is the number of iterations required by the
nonlinear solver to converge for time step n, ml a lower
iteration limit, mu an upper iteration limit, ft a time-step
acceleration factor, Dtmax the maximum allowable timestep size, and Dtmin the minimum allowable time-step
size.
Although this empirical approach can be easily implemented in existing ®xed time-step codes, it requires
the speci®cation of ml , mu , ft , Dtmin , and Dtmax for which
theoretical guidance does not exist. In this work, Dtmin ,
and Dtmax were chosen to yield a set of convergent solutions 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
semi-discrete form is
dpi
A…p†i
ˆ Osdi …p†;
dt

…13†

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 nonsmooth. For simulation of variably saturated ¯ow in
non-uniform media, non-integral interblock permeability 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 expressed19 as
8 R maxfp ;p g
i i1
dw
>
< minfpi ;pi1 g …K dp †dp
R maxfpi ;pi1 g dw
if pi 6ˆ pi1 ;
…15†
Ki1=2
… dp †dp
minfpi ;pi1 g
>
:
if pi ˆ pi1 :
Ki
This approach is not routinely used to solve the untransformed RE due to the apparent computational
expense. However, an integral approach can be used to
solve the transformed RE eciently by taking advantage of tabulation and accurate interpolation methods19.
3.8 Error tolerance
Setting the relative and absolute error tolerances is an
important consideration in using DASPK to solve the
transformed RE. These tolerances are based on the

An evaluation of temporally adaptive transformation approaches for solving Richards' equation
dependent variable, which is p in the transformed case.
For highly nonlinear transforms such as IT2, p values
can be extremely small (10ÿ18 10ÿ6 ) 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 in pi j 6 EWT…i†;

…16†

where
EWT…i† ˆ rtol  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
pi is 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 atol ˆ 0, which
represents a strictly relative error tolerance. By specifying 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 represents a signi®cant portion of the computational e€ort
required to solve RE numerically, due to the complicated 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

835

weights for the three procedures and integer counts for
each of the procedures, such as
…18†
Wn ˆ wj nj ‡ wf nf ‡ wl nl ;
where Wn is 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 untransformed RE using ®nite-di€erence approximation to
evaluate the Jacobians and Hermite spline interpolation
of constitutive relations31: …wj †ut ˆ 0:631, …wf †ut ˆ 0:311,
and …wl †ut ˆ 0:181. 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 †tr ˆ 0:883, …wf †tr ˆ 0:435, and …wl †tr ˆ 0:181. These
weighting factors result in approximately 30% additional 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
"
#1=k
nn
1X
k
^
;
…19†
…jwi ÿ wi j†
jjD jjk ˆ
nn iˆ1
^ is an accurate apwhere k is the norm measure, and w
i
proximation of the true solution based on a dense spatial grid. k ˆ 1, k ˆ 2, and k ˆ 1 were considered in this
work and termed L1 , L2 , and L1 error norms, respectively. 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 performance-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 optimal b value
…bopt † is found by solving the constrained minimization
problem
min

bmin 6 b 6 bmax

fWn …b†  jjD …b†jj1 g;

…20†

where b is the arbitrary transform parameter, bmin and
bmax are the minimum and maximum b values resulting
in convergence of the nonlinear solver, Wn is the required
work as de®ned by eqn (18), and jjD jj1 is the L1 error
norm as de®ned by eqn (19).
For the parameter optimization, we used the nonlinear optimization package IFFCO49. IFFCO is a

836

G. A. Williams, C. T. Miller

projected quasi-Newton algorithm that uses a decreasing sequence of ®nite di€erence steps (scales) to approximate 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 conventional Newton method that an optimal result will be
returned at a local minimum rather than a global minimum.

4 RESULTS AND DISCUSSION
4.1 Test problems
We compared the two temporally adaptive approaches
for solving RE for both the transformed and untransformed 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 discretization intervals. These test problems represent a variety
of media and auxiliary conditions. The material properties 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 Parrish52 from analyses of a large number of soils.
Problems A and C are dicult to simulate numerically due to the relatively dry initial conditions and the
steep wetting fronts that develop. These problems provided 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-

Table 1. Test problem simulation conditions
Variable

Problem A

Problem B

Problem C

hr (±)
hs (±)
av …mÿ1 )
nv (±)
Ks (m/day)
Ss (mÿ1 )
z (m)
t (days)
w0 (m)
w1 (m)
w2 (m)
Dz (m)
Dtmin (days)
Dtmax (days)
ft
ml
mu

0.093
0.301
5.47
4.264
5.040
1:00  106
[0,10.0]
[0,0.18]
ÿz
0.00
0.10
0.0125
1  10ÿ10
1  10ÿ2
1.2
[1,15]
[6,50]

0.102
0.368
3.35
2.000
7.970
0.00
[0,0.3]
[0,0.092]
ÿ10.00
ÿ10.00
ÿ0.75
0.0025
1  10ÿ10
1  10ÿ2
1.2
[1,15]
[5,30]

0.078
0.430
3.60
1.560
0.250
1:00  10ÿ6
[0,5.0]
[0,2.25]
ÿz
0.00
0.10
0.025
1  10ÿ5
2  10ÿ1
1.2
[1,15]
[10,200]

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

tially saturated initially, and fully saturated conditions
do not develop. Yet, Problem B allowed us to compare
our results with recent research performed using stateof-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 performance for four di€erent solution approaches: an
EBATS solution (as outlined in Section 3.5) of the untransformed RE; a DASPK solution of the untransformed RE; an EBATS solution of the transformed 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 produced 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, respectively; 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 simulations; 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

An evaluation of temporally adaptive transformation approaches for solving Richards' equation

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

837

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 approach for intermediate to high levels of accuracy;
(ii) transformed DASPK solutions tended to be increasingly ecient compared to untransformed
DASPK solutions as nv increased, 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-order 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 performance of this solution approach will depend on the IT2
parameter b.
4.3 Parameter sensitivity

Fig. 3. Error vs. work comparisons for Problem B.

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

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

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 ; and nv 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
‡
as bÿ
90 and b90 , 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 conditions did not develop, bopt was relatively insensitive
to changes in Dz; av ; and nv ;
(ii) in general, bopt was more sensitive to Dz in regions
of relatively ®ne or coarse spatial discretizations and
less sensitive in regions of intermediate spatial discretization scales.
(iii) for non-uniform media …nv 6 2:0†; bopt approached zero and was relatively insensitive to changes in media parameters av and nv ;

838

G. A. Williams, C. T. Miller

Table 2. Optimal Parameter Range vs. Spatial Discretization
Dz …10ÿ2 †

Problem A

0.98
1.25
1.56
1.95
2.50
3.13
3.75
4.69
5.00
6.25
7.81
9.38
10.00
12.50
20.00

Problem B

bÿ
90

bopt

b‡
90

0.080
0.050
0.050
0.040
0.040
0.030

0.100
0.080
0.070
0.060
0.050
0.060

0.450
0.140
0.100
0.120
0.100
0.110

0.050
0.030
0.040

0.060
0.050
0.050

0.100
0.090
0.090

0.005
0.001
0.001

0.009
0.002
0.001

0.010
0.008
0.006

Problem C

bÿ
90

bopt

b‡
90

0.001

0.003

0.006

0.003
0.001
0.002
0.001
0.001
0.002

0.006
0.002
0.003
0.001
0.002
0.003

0.008
0.005
0.005
0.007
0.006
0.006

0.001
0.001

0.002
0.002

0.006
0.005

bÿ
90

bopt

b‡
90

0.030
0.036
0.007
0.003
0.001
0.001

0.065
0.044
0.010
0.006
0.004
0.002

0.100
0.060
0.015
0.007
0.006
0.003

0.001
0.010
0.004

0.001
0.013
0.005

0.002
0.017
0.008

0.002
0.001
0.001

0.002
0.001
0.001

0.005
0.003
0.003

Table 3. Optimal Parameter Range vs. av
Problem A

av

2.0
5.0
8.0
1.0
3.0

Problem B

Problem C

bÿ
90

bopt

b‡
90

bÿ
90

bopt

b‡
90

bÿ
90

bopt

b‡
90

0.030
0.040
0.015
0.017
0.046

0.060
0.050
0.020
0.020
0.050

0.075
0.085
0.030
0.033
0.080

0.001
0.003
0.001
0.001
0.001

0.001
0.004
0.003
0.001
0.002

0.002
0.008
0.008
0.006
0.007

0.001
0.001
0.008
0.002
0.002

0.001
0.001
0.008
0.004
0.002

0.002
0.002
0.010
0.005
0.005

Table 4. Optimal Parameter Range vs. nv
Problem A

nv

1.5
2.0
2.6
3.2
3.8
4.4

Problem B

Problem C

bÿ
90

bopt

bÿ
90

bÿ
90

bopt

b‡
90

bÿ
90

bopt

b‡
90

0.001
0.004
0.080
0.170
0.210
0.070

0.001
0.006
0.110
0.180
0.310
0.090

0.002
0.007
0.140
0.185
0.390
0.150

0.001
0.003
0.001
0.001
0.001
0.001

0.001
0.006
0.001
0.006
0.001
0.001

0.002
0.010
0.007
0.010
0.006
0.009

0.001
0.001
0.001
0.002
0.001
0.001

0.001
0.001
0.002
0.003
0.001
0.001

0.002
0.002
0.003
0.005
0.002
0.002

(iv) for dicult, initially dry, sharp-front problems
such as A, bopt appeared to be more sensitive to variations in nv than to variations in av ; and
(v) the performance of the DASPK solver, as measured by the objective function work  error, was less
sensitive to small deviations form bopt than the performance of ®xed time-step solvers.
4.4 Parameter estimation
The parameter sensitivity data did not produce a wellde®ned estimate of the optimal transform parameter
that was reliable over the entire range of media and
spatial discretization parameters. It did, however, provide some insight for guiding selection of transform

parameters for several classes of problems. For problems involving low nv values … 6 2:0† or where saturated
conditions do not develop, small b 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 complicated. Based on our numerical experiments, it appears
that an estimator for these types of problems must involve some function of Dz and nv . 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

An evaluation of temporally adaptive transformation approaches for solving Richards' equation
near-optimalperformance is acceptable, b values within
the 0.01±0.3 range will usually produce acceptable results for these types of problems. This near-optimal
performance is usually more ecient than that of alternative solution strategies, such as untransformed,
®xed time-step, or empirically based adaptive timestepping.
5 CONCLUSIONS
We can draw several conclusions based on our work.
· We outline a set of computational techniques ± including an MOL formulation, a recently introduced 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 robust 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 empirically 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 intermediate to high levels of accuracy.
· The potential bene®t of a particular transform depends 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 performance to deviations from the optimal parameter.
ACKNOWLEDGEMENTS
This work supported in part by US Army Waterways
Experiment Station Contract DACA39-95-K-0098, Army Research Oce Grant DAAL03-92-G-0111, National Institute of Environmental Health Sciences Grant
5 P42 ES05948, and a Department of Energy Computational Science Fellowship. Computing activity was
partially supported by allocations from the North Carolina Supercomputing Center.
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