HESSHydrology and Earth System SciencesHESSHydrol. Earth Syst. Sci.1607-7938Copernicus PublicationsGöttingen, Germany10.5194/hess-21-2667-2017Ross scheme, Newton–Raphson iterative methods and time-stepping strategies
for solving the mixed form of Richards' equationHassane MainaFadjiAckererPhilippeackerer@unistra.frLaboratoire d'Hydrologie et de Géochimie de Strasbourg, Univ. Strasbourg/EOST – CNRS, 1 rue Blessig, 67084 Strasbourg, FranceCEA-Laboratoire de Modélisation des Transferts dans
l'Environnement, Bât. 225, 13108 Saint Paul lez Durance cedex, FrancePhilippe Ackerer (ackerer@unistra.fr)8June20172162667268323November201620December20165May20178May2017This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://hess.copernicus.org/articles/21/2667/2017/hess-21-2667-2017.htmlThe full text article is available as a PDF file from https://hess.copernicus.org/articles/21/2667/2017/hess-21-2667-2017.pdf
The solution of the mathematical model for flow in variably
saturated porous media described by the Richards equation (RE) is subject to
heavy numerical difficulties due to its highly nonlinear properties and
remains very challenging. Two different algorithms are used in this work to
solve the mixed form of RE: the traditional iterative algorithm and a
time-adaptive algorithm consisting of changing the time-step magnitude within
the iteration procedure while the nonlinear parameters are computed with the
state variable at the previous time. The Ross method is an example of this
type of scheme, and we show that it is equivalent to the Newton–Raphson
method with a time-adaptive algorithm.
Both algorithms are coupled to different time-stepping strategies: the
standard heuristic approach based on the number of iterations and two
strategies based on the time truncation error or on the change in water
saturation. Three different test cases are used to evaluate the efficiency of
these algorithms.
The numerical results highlight the necessity of implementing an estimate of
the time truncation errors.
Introduction
Water movement in soils is one of the key processes in the water cycle since
it contributes to the renewal of groundwater resources through recharge, to
vegetation growth through transpiration, to soil fertility through
salinization/alteration and to atmospheric humidity through evaporation and
transpiration. Water movement is usually modeled using the Richards equation
(Richards, 1931), which is now commonly adopted for many studies in soil
science and/or hydrology, including the use of physically based hydrological
models applied to large-scale catchments and for long time simulations (e.g.,
for climate change studies). However, this equation is highly nonlinear, and
despite numerous efforts over the last 40 years, its numerical solution
requires much computational time.
Assuming a rigid solid matrix, the Richards equation (RE) is given
by
∂θ∂t+Sws0∂ψ∂t+∇⋅q=fq=-kr(ψ)K∇ψ+∇z
where θ is the volumetric water content
(L3 L-3), Sw is the water saturation (–), s0 accounts
for fluid compressibility (L-1), ψ is the pressure head (L), q
is the water flux based on the extended Darcy's law (L T-1), t is the
time (T), z is the vertical coordinate (positive upward) (L), f is the
sink/source term (T-1), K is the saturated hydraulic conductivity
tensor (L T-1) and kr(ψ) is the relative hydraulic
conductivity (-). The model includes initial and boundary conditions of the
Dirichlet (prescribed pressure head) or Neumann (prescribed flux) type.
Equation (1) is also called the mixed form of RE. Two alternative
formulations of the mixed form exist for RE.
The pressure form is defined by
Cψ+Sws0∂ψ∂t+∇⋅q=fq=-kr(ψ)K∇ψ+∇z
where Cψ=∂θ∂ψ is the
specific moisture capacity (L-1), and the soil moisture form that is
restricted to unsaturated conditions is defined by
∂θ∂t+∇⋅q=fq=-D(θ)∇θ+kr(θ)K∇z
where D(θ)=kr(θ)Kdψdθ is the
pore water diffusivity (L2 T-1).
Constitutive relations are required to solve RE. For the pressure–water
content relationship, the most common model is the Van Genuchten model (van
Genuchten, 1980):
Sw(ψ)=θ(ψ)-θrθs-θr=1+αψη-mψ<0,1ψ≥0,
where m=1-1/η, Sw is the effective saturation,
θr and θs are the residual and saturated
volumetric water content, respectively, and α and η are
experimentally estimated coefficients.
This model is usually associated with the Mualem model (Mualem, 1976) for the
relative permeability of the aqueous phase:
kr(Sw)=Sw1/21-1-Sw1/mm2ψ<0,1.0ψ≥0.
A summary of the most popular relations can be found in
Belfort et al. (2013).
Due to the strong heterogeneities of the unsaturated zone and nonlinearities
in the constitutive relations (Eqs. 4 and 5), analytical solution of RE
does not exist except in special cases
(Celia et al., 1990; van Dam and Feddes,
2000). Therefore, numerical methods such as finite difference
(Feddes
et al., 1988; Romano et al., 1998; van Dam and Feddes, 2000), finite element
(Gottardi and Venutelli, 2001), and mixed
finite element
(Bause
and Knabner, 2004; Bergamaschi and Putti, 1999; Fahs et al., 2009; Farthing
et al., 2003) are used to solve RE.
Iterative methods based on the Picard (fixed point) or Newton–Raphson
approach (Lehmann and Ackerer, 1998; Paniconi and Putti, 1994) are the most
popular techniques for solving this highly nonlinear equation. Alternative
iterative methods are based on transform formulations (Crevoisier et al.,
2009; Ross and Bristow, 1990; Williams et
al., 2000; Zha et al., 2013) or the method of lines (Fahs et al., 2009;
Matthews et al., 2004; Miller et al., 1998; Tocci et al., 1997).
Adaptive time-stepping strategies based on time truncation error control were
found to be superior to other approaches (Hirthe and Graf, 2012; Kavetski
et al., 2001; Tocci et al., 1997). The method of lines using the DASPK
integrator was applied to the Richards' equation by Matthews et al. (2004),
Miller et al. (1998), and Tocci et al. (1997), among others. The method of
lines consists of discretization of the spatial part of the PDE only, leading
to a system of ordinary differential equations. It has been found to be
significantly more efficient than other temporal discretizations (Miller et
al., 2006). However, Kavetski and Binning (2002b) reported difficulties in
obtaining convergence for the DASPK solver associated with an arithmetic mean
of inter-block conductivities for the most difficult problem addressed by
Miller et al. (1998). Additionally, very few non-iterative schemes have been
developed (Kavetski and Binning, 2004, 2002a; Paniconi et al., 1991).
Despite the many existing numerical methods, solution of RE is still a
challenging research topic, with many remaining questions about reduction of
the computational time, treatment of nonlinearities, and improvement of the
accuracy of these methods for difficult problems such as infiltration in very
dry soils (Diersch and Perrochet, 1999; Forsyth et al., 1995; Hills, 1989).
The need for efficient algorithms to solve this equation has increased during
recent decades because it has been recognized that explicit
modeling of flow in the unsaturated zone
has to be implemented in land surface models (Vergnes et al., 2012). In their
recent review of land surface models, Clarke et al. (2015) push for a
mechanistic modeling of the flow in soils. They consider the implementation
of the mixed form of the Richards equation to be an improvement to the
modeling of soil moisture variations. They also underline the need for
efficient algorithms to solve RE, to allow the implementation of stochastic
approaches and/or automatic parameter estimations.
In this study, we analyzed the performance of different algorithms based on
the Newton–Raphson method since the classical Picard scheme has been found
to be less efficient (Lehmann and Ackerer, 1998). Applied to the soil
moisture form of RE, we demonstrate that the recently developed Ross method
(Ross, 2003; Crevoisier et al., 2009; Zha et al., 2013) is equivalent to the
Newton–Raphson method (Sect. 2). A detailed presentation of the
Newton–Raphson method applied to the mixed form or RE is given in Sect. 3.
The standard Newton–Raphson algorithm is based on the computation of the
corresponding matrices in an iterative way by updating the parameters until
convergence. An alternative algorithm has been suggested more recently where
the parameters are kept unchanged within one time step and the time step is
adapted to reach convergence. This algorithm has been applied to the
pressure-based form of RE by Kavetski and Binning (2002a) and to the soil
moisture form by Crevoisier et al. (2009), Ross (2003), and Zha et
al. (2013). Although this algorithm is called “non-iterative” because the
parameters are not updated during the calculation, iterations may be
necessary to adapt the magnitude of the time step. Therefore, in the
following, we will refer to the usual algorithm as “iterative” and to the
alternative algorithm as “time-adaptive”. To our knowledge, this
alternative algorithm has never been applied to the mixed form of RE.
Section 4 is dedicated to both algorithms and to the time-stepping strategy
used for solving RE. Finally, in Sect. 5, the numerical accuracy and
robustness of the algorithms applied to the mixed form of RE are evaluated
using three different test cases.
The Ross method and the Newton–Raphson method
The moisture-based formulation is applicable in unsaturated conditions only
and is prone to numerical difficulties in the case of heterogeneous soils,
explaining the reduced attention directed to this formulation. However,
discontinuous water content can be handled by adapted schemes, and the
moisture-based formulation appears to be very accurate for initially dry
conditions (Zha et al., 2013, 2015).
Ross (2003) suggested a non-iterative formulation that has been
recently extended to different soil conditions
(Crevoisier
et al., 2009; Varado et al., 2006a) and to two and three dimensions
(Zha et al., 2013).
In its initial one-dimensional finite-volume formulation and for a volume
(cell) i, the Ross method (Ross, 2003) is based on the following
set of equations:
ΔzΔtθin+1-θin=ΔzΔtθs,i-θr,iSin+1-Sin=q-σ-q+σ
with
q+σ=q+n+σ∂qin∂SinSin+1-Sin+∂qin∂Si+1nSi+1n+1-Si+1nq-σ=q-n+σ∂qin∂SinSin+1-Sin+∂qin∂Si-1nSi-1n+1-Si-1n,
where Sin+1 is the water saturation at cell/node i at time
(n+1), q-σ (or q+σ) is the water flux
between cell i and (i-1) (or i+1) at time t=tn+σΔt,σ∈0,1 and Δz is the size of cell i.
θs,i is the saturated water content and θr,i is the
residual water content. For simplicity, we assume here that all cells are of
the same size.
The previous mass balance Eq. (6) leads to the following equation for
cell i:
-∂q-n∂Si-1nSi-1n+1-Si-1n+ΔzσΔtθs,i-θr,i-∂q-n∂Sin-∂q+n∂SinSin+1-Sin+∂q+n∂Si+1nSi+1n+1-Si+1n=q-n-q+n.
The Newton–Raphson method was initially developed as a root-finding
algorithm of an arbitrary equation that has been generalized for solving a
system of nonlinear equations. Applied to the soil moisture form of RE and
using an implicit scheme, the NR consists in defining a residual based on the
mass balance equation (Eq. 6) at iteration k for time step n+1 and for
cell i written as
Rin+1,k=ΔzΔtθs,i-θr,iSin+1,k-Sin+q+n+1,k-q-n+1,k,
where Rin+1,k is called the residual.
The NR consists in computing the solution at iteration k+1 by estimating
the residual of the next iteration Rin+1,k+1 using a
first-order Taylor development and setting it equal to zero as
Rin+1,k∂Sn+1,kSin+1,k+1-Sin+1,k+Rin+1,k=0.
The derivatives of this residual are
∂Rin+1,k∂Si-1n+1,k=-∂q-n+1,k∂Si-1n+1,k,∂Rin+1,k∂Sin+1,k=ΔzΔtθs,i-θr,i+∂q+n+1,k∂Sin+1,k-∂q-n+1,k∂Sin+1,k,∂Rin+1,k∂Si+1n+1,k=∂q+n+1,k∂Si+1n+1,k,
which leads to the following set of linear equations:
-∂q-n+1,k∂Si-1n+1,kSi-1n+1,k+1-Si-1n+1,k+ΔzΔtθs,i-θr,i+∂q+n+1,k∂Sin+1,k-∂q-n+1,k∂Sin+1,kSin+1,k+1-Sin+1,k+∂q+n+1,k∂Si+1n+1,kSi+1n+1,k+1-Si+1n+1,k=ΔzΔtθs,i-θr,iSin+1,k-Sin+q+n+1,k-q-n+1,k.
For the first iteration, we have Sin+1,k+1=Sin+1 and
Sin+1,k=Sin, and therefore
-∂q-n∂Si-1nSi-1n+1-Si-1n+ΔzΔtθs,i-θr,i+∂q+n∂Sin-∂q-n∂SinSin+1-Sin+∂q+n∂Si+1nSi+1n+1-Si+1n=q+n-q-n.
Whatever the formulation of the fluxes q (as a function of the pressure
(see Eq. A1) or the water content, expressed by Kirchhoff transform as in
Ross, 2003, or not), the implicit Ross method (Eq. 8 with σ=1) is
equivalent to the first iteration of the Newton–Raphson method (Eq. 13).
Newton–Raphson method for the mixed form Richards' equation
Because the pressure-based formulation does not ensure mass conservation –
except for the approximation provided by Rathfelder and
Abriola (1994) – and due to the limitations of the moisture-based
formulation (see previous section), the mixed formulation has been widely
used since the work of Celia et al. (1990).
The mixed form of the Richards equation given by Eq. (1) is rewritten as
∂θ∂t+Sws0∂ψ∂t=∇⋅kr(ψ)K∇ψ+∇z+f
and is discretized by
An+1,kψn+1,k+1+Bn+1,kψn+1,k+1-ψnΔtn+1+Eθn+1,k+1-θnΔtn+1=Fn+1,k,
where A is the discretized form of the divergence term,
B and E are the discretized forms of the storage terms,
F is the discretized form of the sink/source term and the boundary
conditions, n is the time step and k is the iteration counter. Δtn+1 is the time-step magnitude defined by Δtn+1=tn+1-tn. Matrices A, B, and E
and vector F depend on the numerical scheme used for the spatial
discretization. The implicit scheme is applied for the spatial
discretization.
For the Newton–Raphson method, the residual is defined now by
R(ψn+1,k)=An+1,kψn+1,k+Bn+1,kψn+1,k-ψnΔtn+1+Eθn+1,k-θnΔtn+1-Fn+1,k
and its derivatives are
Looking for ψn+1,k+1 such as
R(ψn+1,k+1)=0, the system to be solved is similar to
Eq. (10):
R′(ψn+1,k)Δψn+1,k+1=-R(ψn+1,k),
with Δψn+1,k+1=ψn+1,k+1-ψn+1,k.
The NR formulation is also used for the non-iterative scheme by applying
only one NR step per time step, with ψn+1=ψn+1,1
where ψn+1,0=ψn
(Paniconi et al., 1991; Zha et al.,
2015).
Algorithms and time-stepping strategy
The usual algorithm used to solve RE consists in defining a time step that
remains constant and in iteratively computing the parameters and variables in
the following way.For a given time step n
Define the time-step length Δtn+1 depending on the time-stepping
strategy.
Initialization of the iterative process by setting ψn+1,1=ψn.
do k=1, maxit
Computation of the variable θn+1,k, the parameter
Kn+1,k and their derivatives dθn+1,kdψn+1,k, ∂Kn+1,k∂ψn+1,k using ψn+1,k.
Computation of the system matrix R′ and the residual R.
Computation of the system solution ψn+1,k+1.
Check convergence. If convergence is achieved, exit.
enddo
Next time step
where k is the iteration counter and maxit the maximum number of iterations.
The time-adaptive algorithm consists in calculating the nonlinear parameters
with the pressure heads computed at time step n and adapting the time-step
length. The algorithm is described by the following.
For a given time step n
Computation of the variable θn, the parameter
Kn and their derivatives dθndψn, ∂Kn∂ψn using ψn.
do k=1, maxit
Define a time step Δtn+1,k depending on the time-stepping
strategy.
Computation of the system matrix R′ and the residual R.
Computation of the system solution ψn+1,k+1.
Check convergence. If convergence is achieved, exit.
enddo
Next time step
The main advantage of the alternative algorithm is its avoidance of the
computation of the variable θ, the parameter K and
their derivatives dθdψ and ∂K∂ψ during the iterations. Due to the highly
nonlinear relations between θ, K, dθdψ, ∂K∂ψ and the
pressure, this computation may require significant CPU time.
The most popular time-step management during the simulation is that of the
heuristic type (Miller et al., 2006). The time step Δtn+1 is
computed depending on Δtn and the number of iterations k
necessary to reach convergence in the following way:
ifk≤m1Δtn+1=k1Δtnk1>1.0,ifm1≤k≤m2Δtn+1=Δtn,ifm2≤kΔtn+1=k2Δtnk2<1.0,
where k1, k2, m1, and m2 are user-defined constants.
Other heuristic time-step management procedures have been suggested by
Kirkland et al. (1992) based on the water volumes exchanged between the
adjacent cells of the grid, and by Ross (2003), where the time-step size is
controlled by the maximum allowed change in the saturation.
For the Ross method, the fluxes are computed first and the time-step
magnitude is calculated accordingly using
Δtn+1=ΔSmaxmaxiq-,in-q+,inΔziθs,i-θr,i
where ΔSmax is the user-defined maximum allowed saturation
change. After the computation of the change in the saturation ΔS, the
time step is modified if the maximum of the computed change exceeds 1+λmaxiΔSi, where λ is a user-defined value, according to
Δtn+1,k=ΔSmaxmaxiΔSiΔtn+1,k-1,
and the system of equations is solved again. More details about handling the
fluxes at boundaries and saturated conditions can be found in
Crevoisier
et al. (2009), Ross (2003) and Varado et al. (2006b).
The adaptive scheme used in this work evaluates the time steps through
truncation error due to the temporal discretization as proposed by
Thomas and Gladwell (1988). This scheme was already applied to
the pressure-based formulation by Kavetski et al. (2001) and to the moisture-based formulation by Kavetski and
Binning (2004).
The difference between the first-order and second-order time approximations
can be considered as an estimate of the local truncation error of the
first-order scheme. The first-order approximation is given by
ψ1n+1=ψn+Δtn+1∂ψn∂t.
The second-order approximation is
ψ2n+1=ψn+Δtn+1∂ψn∂t+12Δtn+12∂2ψn∂t2=ψn+12Δtn+1∂ψn+1∂t+∂ψn∂t
using ∂ψn+1∂t=∂ψn∂t+Δtn+1∂2ψn∂t2.
This truncation error is given by
εtn+1=maxiψ2,in+1-ψ1,in+1=12Δtn+1maxi∂ψin+1∂t-∂ψin∂t≈12Δtn+1maxiψin+1-ψinΔtn+1-ψin-ψin-1Δtn,
when the truncation error is smaller than γ, the temporal truncation
error tolerance defined by the user, and the size of the next time step
calculated by
Δtn+1=Δtnminsγmax(εtn+1,EPS),rmax.
When the truncation error is larger than γ, the computation is
repeated with a reduced time step defined as follows:
Δtn=Δtnmaxsγmax(εtn+1,EPS),rmin,
where rmax and rmin are user-defined constants used
to avoid overly drastic changes in the time step. s is considered to be a
safety factor that ensures that the time-step changes are reasonable. EPS is
used to avoid floating point errors when the truncation error becomes too
small.
Evaluation of the algorithms' performance
We applied the NR method to the mixed form of RE using the standard iterative
algorithm and the time-adaptive algorithm. A cell centered finite-volume
scheme for the spatial discretization with an implicit Euler scheme for the
temporal discretization has been used to solve the partial differential
equation and arithmetic means are used to compute the inter-block hydraulic
conductivity. The detailed discretizations of the matrix
R′(ψn+1,k) and the vector R(ψn+1,k)
(see Eq. 18) are given in Appendix A. The time-adaptive algorithms have been
applied as described by the authors: Ross (2003) for the time stepping based
on the saturation changes and Kavetski et al. (2001) for the time stepping
based on the truncation errors.
Different options of the tested algorithms. Reference to the
corresponding equation in parentheses.
For the standard iterative algorithm, we defined two types of errors to check
the convergence: the error based on the maximum change in the state variables
between two iterations defined by εψ=maxiψin+1,k+1-ψin+1,k and the truncation error
εt defined by Eq. (24). Convergence is assumed to be achieved
when
εψ<τψ,a+τψ,rψimaxn+1,k+1,
where τψ,a and τψ,r are the absolute and relative
user-defined tolerances and ψimaxn+1,k+1 is the pressure
corresponding to εψ and when
εt<τt,a+τt,rψimaxn+1,k+1,
where τt,a and τt,r have the same meaning as those for
the previous criterion but ψimaxn+1,k+1 represents the pressure
value corresponding to εt.
The tested algorithms are summarized in Table 1. Computations of all possible
combinations for the standard iterative scheme have been performed. We
present only the four most efficient algorithms. We also analyzed convergence
based on the nonlinear residual. It was found to be less restrictive than the
previous criteria. Due to the definition of the NR method, the residual tends
to zero, but it does not ensure a small value of εψ.
Therefore, the results related to the reduction of the
nonlinear residuals are not reported.
Domain size (L), initial conditions (IC), boundary conditions at
the soil surface (BCu) and at the soil bottom (BCl), saturated
hydraulic conductivity (Ks), residual and saturated water
contents (θr, θs) and shape parameters
(α, η) for the different test cases. KM(t) is the
hydraulic conductivity of the last grid cell. Length and time units are
centimeters and seconds, respectively.
Relative permeability as a function of the pressure for the three
test cases (L1, L2 and L3 are the three layers for test case 3).
We investigated three one-dimensional problems with various initial and
boundary conditions and hydraulic functions to assess the accuracy,
efficiency and computational costs of the different algorithms. The selected
test cases represent a range of difficult infiltration problems widely
analyzed in the literature.
TC1: infiltration in a homogeneous initially dry soil with constant
prescribed pressure at the surface and prescribed pressure at the bottom
(Celia et al., 1990);
TC2: infiltration in a homogeneous soil initially at hydrostatic equilibrium
with a prescribed constant flux at the soil surface and prescribed pressure
at the bottom (Miller et al., 1998); and
TC3: infiltration/evaporation in an initially dry heterogeneous soil,
with
variable positive and negative fluxes at the surface and free drainage at
the base of the soil column (Lehmann and Ackerer,
1998).
For the three test cases, the soil hydraulic functions were described by
Mualem–Van Genuchten models (Mualem, 1976; van Genuchten, 1980); see
Eqs. (4) and (5).
The required parameters, boundary conditions and initial conditions are
summarized in Table 2. The evolution of the relative hydraulic conductivity,
the water saturation and the specific moisture capacity with respect to the
pressure values are shown in Figs. 1, 2 and 3, respectively. For TC1, the
pressure will vary from -1000 to -75 cm only due to the specific
conditions of this test case. Therefore, the parameter variations are
smaller than those for the other test cases. Since the parameters'
variations are more abrupt for test cases 2 and 3, their solutions are more
challenging.
Water saturation as a function of the pressure for the three test
cases (L1, L2 and L3 are the three layers for test case 3).
Preliminary tests were performed to define the optimal spatial
discretization; i.e., a finer spatial discretization provided very similar
results for a given convergence criterion and a given time-stepping strategy.
Therefore, we can assume that the errors only originate from the time-step
size and the linearization.
The following criteria were used for the time-stepping strategy:
k1= 0.80, k2= 1.20, m1= 5, and m2= 10, which are the usual
values for the heuristic strategy defined by Eq. (19); and
rmin= 0.10, rmax= 4.0, s= 0.9, and EPS = 10-10, which are the
standard values for the time-stepping scheme based on the time discretization
error defined by Eq. (26) (Kavetski et al., 2001).
To perform a consistent comparison of the time-stepping strategies, the
maximum allowed change in saturation (see Eqs. 20 and 21) has been evaluated
using the maximum change in the pressure, according to the following
relationship:
ΔSmax≈1θs,imax-θr,imaxdθdψimaxnτa+τrψimaxn+1,k+1.
The simulations have been performed using different values of τr
and with τa=0.0.
Specific moisture capacity as a function of the pressure for the
three test cases (L1, L2 and L3 are the three layers for test case 3).
We used several criteria to evaluate the performance of these codes. A
typical error used in solving RE is the global cumulative mass balance error
defined by
MB(tn+1)=∑i=1MΔziθin+1-θi0∑k=1n+1qink-qoutkΔtk,
where Δzi is the size of the cell/element i, θin+1 is its water content at time tn+1, θi0 is the
initial water content, and qink and qoutk are
the inflow and outflow, respectively, at the domain boundaries at time
tk. M is the number of cells/elements. The fluxes at the boundaries
are defined by qk=12qk+qk-1. The mass
balance errors were checked for each run but were found to be negligible
since we solved the mass-conserving RE form.
While it is necessary to satisfy the global mass balance for an accurate
numerical scheme, a low mass balance error is not sufficient to ensure the
accuracy of the solution. Therefore, solutions have also been compared with
the reference solution obtained using a very fine temporal discretization and
the iterative Newton–Raphson method. This comparison is based on the average
relative error defined by
εk=1M∑iψiref-ψ^ikψirefk1/k,
where M is the number of cells, ψref is the
reference solution and ψ^ is the tested numerical
solution. ε1 represents the average absolute relative error
(called L1-norm in the following), ε2 is the average
quadratic error (L2-norm) and ε∞ is the highest
local relative difference between the numerical and reference solutions
(L∝-norm).
Relative errors and number of iterations obtained for the iterative
algorithm depending on different convergence criteria for TC1.
Since the time-adaptive algorithm does not require the computation of the
parameters and their derivatives during the iterative procedure, we use
Nsol to denote the number of times where the system of equations
is solved and Nparam to denote the number of times where the
parameters are computed. Of course, these counters are equal to each other
for the standard algorithm, which leads to computational costs depending on
2Nsol. Nparam is less than Nsol for the
time-adaptive algorithm. For comparison purposes, the computational costs are
estimated by Nsol for the standard algorithm and by
(Nsol+Nparam)/2 for the time-adaptive algorithm. The
efficiency of the algorithms has been evaluated by comparing the
computational costs for a given relative tolerance τr. The
errors are presented in the tables and the figures. The figures show some
additional results not listed in the tables that already contain much
information.
TC1: Infiltration in a homogenous soil with constant boundary conditions
This test case simulates an infiltration into a homogeneous porous medium.
This problem is addressed here because it has been widely analyzed previously
by many authors like Bouchemella et al. (2015), Celia et al. (1990), El Kadi
and Ling (1993), Rathfelder and Abriola (1994), and Tocci et al. (1997),
among others. The computations were performed with a spatial discretization
of 0.1 cm. The initial time-step size was set to
1.0 × 10-5 s, and the maximum time-step size was set to
400 s.
The results for the iterative and time-adaptive algorithms are presented in
Tables 3 and 4, respectively. When both convergence criteria are used
(algorithms SH_Δψ_Δt and SS_Δψ_Δt), Ntrunc represents the number of times where the truncation
error is the most restrictive condition. For the heuristic time-stepping
schemes, the convergence is mostly linked to the truncation error
(Ntrunc is close to Nsol), whereas when the
saturation time-stepping scheme is used, the most restrictive criterion is
the maximum difference in the pressure.
When the time-stepping scheme is based on saturation, for both iterative and
time-adaptive algorithms, the number of iterations required to solve the
problem is proportional to the relative tolerance. Therefore, highly accurate
solutions incur high computational costs.
For the time-adaptive scheme, the number of parameter changes
Nparam is close to the number of iterations for low tolerance
values. Small tolerance values lead to small time steps, avoiding time-step
adjustments. This is not the case for larger tolerance values that lead to
larger time steps and therefore to additional iterations (see for example
TA_T for the tolerance of τr= 10-2 – Table 4).
Evolution of the L2 relative error with computational costs
for TC1.
The three types of errors provide the same information. The best solution for
one type of error is also the best solution for the other two.
On average, the iterative algorithm is faster than the time-adaptive
algorithm that requires more iterations for a given error. This is also
shown in Fig. 4 that presents the convergence rate of the L2-norm
with respect to the computational costs, i.e., the number of iterations or number
of iterations and number of parameter changes. The time-adaptive algorithm
with time stepping based on the truncation errors performs quite poorly
compared to the other algorithms. Irrespective of the tolerance, this
algorithm leads to a wetting front moving faster (Fig. 5).
When the relative tolerance is set to a very low value
(τr= 10-5), the iterative scheme with time stepping
based on the saturation changes shows behavior that is different from that
found for the less restrictive tolerance. The criterion based on truncation
errors is no longer significant (Ntrunc= 252), possibly
explaining why the accuracy of the scheme remains constant. This also
indicates that errors due to time discretization have to be handled, either
in the convergence criterion or in the time-stepping strategy.
Relative errors and number of iterations obtained for the iterative
algorithm depending on different convergence criteria for TC2 (n.c.: non
convergence in less than 107 iterations).
Pressure profiles in the domain for the TA_T
algorithm.
For this test case, the most efficient algorithms are the iterative
algorithms using the time-stepping strategy based on truncation error
(ST_Δψ) or based on the saturation changes (SS_Δψ_Δt). Saturation-based time-stepping strategies (SS_Δψ_Δt and TA_S) show a linear decrease in L2 with
computational costs. For very high precision (L2 < 10-4),
ST_Δψ outperforms the other algorithms. No convincing
explanation has been found for the insignificant change in accuracy for
SS_Δψ_Δt at high precision.
TC2: Infiltration in a homogenous soil with hydrostatic initial conditions
This test case models an infiltration in a 200 cm vertical column of
unconsolidated clay loam with non-uniform grain size distribution and was
considered by Miller et al. (1998) to be a very challenging test. This
problem was found to be more challenging from the numerical point of view
compared to TC1 due to the relative permeability function that enhances the
nonlinear behavior of Richards' equation (Figs. 1, 2, and 3). The cell size
has been set to 0.125 cm, the initial time step to 10-5 s and the
maximum time-step magnitude to 1000 s.
The different norms for the iterative and time-adaptive schemes are given in
Tables 5 and 6.
Investigation of this test case leads to similar qualitative conclusions when
the time-stepping scheme is based on the saturation differences (SS_Δψ_Δt and TA_S). The standard scheme SH_Δψ fails
to provide an accurate solution within a reasonable number of iterations
(less than 107).
Evolution of the L2 relative error with computational costs
for TC2.
Time-step magnitudes during the simulation for TC2.
The most efficient methods are the schemes using the time-stepping strategy
based on truncation errors (Fig. 6). However, as found for TC1, the adaptive
time algorithm TA_T failed to provide highly accurate results (L2-norm
error less than approximately 4.5 × 10-4).
Figure 7 shows the time-step magnitudes for approximately equal L2-norms
for the two time-adaptive algorithms and for the iterative algorithm using
truncation errors for time stepping (4.254 × 10-4 within 3503
iterations for ST_Δψ, 4.563 × 10-4 within 3098
iterations for TA_T and 4.844 × 10-4 within 11358 iterations
for TA_S). The time-step evolution is very similar for the three strategies:
a linear increase until around 0.1 s, followed by a very slow increase until
20–30 s and a regular increase until the end of the simulation. ST_Δψ and TA_T strategies lead to the same time steps when the time reaches
1 s. The time-step sizes remain smaller for TA_S, which explains the
significantly higher number of iterations required to solve this test case.
TC3: Infiltration/evaporation in a heterogeneous soil
This case study simulates infiltration in an initially dry heterogeneous soil
with a succession of rainfall and evaporations as upper boundary conditions
during 35 days. This problem differs from the two previous cases by the soil
heterogeneity and also by the non-monotonic boundary conditions at the soil
surface. It is expected that non-monotonic discontinuous boundary conditions
will increase the difficulty in finding accurate solutions. The soil profile
consists of three 60 cm thick layers. The layers are discretized using cells
with the size of 0.10 cm. The prescribed fluxes are changing every day. For
a given time, these fluxes are linearly interpolated. To avoid an overly
rough time discretization of these boundary conditions, the maximum time-step
magnitude has been fixed at 0.20 days. The initial time step is set to
10-5 days.
The relative errors estimated by the iterative algorithms and the
time-adaptive algorithms are presented in Tables 7 and 8, respectively, and
are plotted in Fig. 8.
Relative errors and number of iterations obtained for the iterative
algorithm depending on different convergence criteria for TC3 (n.c.: non
convergence in less than 107 iterations, * convergence failed for
10-3, τr= 0.90 × 10-3).
Evolution of the L2 relative error with computational costs
for TC3.
Time-step magnitudes during the simulation for TC3 for the
time-stepping strategy based on truncation error (TA_S in blue, TA_T in
black, time-varying boundary conditions at the top).
The standard iterative scheme fails to converge within the maximum number of
iterations (107) when the tolerance is not sufficiently restrictive. The
detailed analyses of the computation showed that the time-step size was quite
large compared to the more restrictive conditions until day 28.0, where the
infiltration fluxes were equal to 1.50 cm day-1 and where the
conditions were near saturation due to the previous infiltration period. This
led to a decrease in the time step to close to the minimum value
(10-8 s), causing the procedure to stop. More restrictive conditions
lead to smaller time steps from the beginning of the simulation and a better
approximation of the solutions during the entire simulation.
The iterative scheme coupled with the truncation-based time-stepping strategy
showed surprisingly unstable behavior for τr= 10-3. The
scheme did not converge for τr∈0.96×10-3;1.04×10-3. The results
presented in Table 7 and Fig. 8 are obtained for τr=0.90×10-3. At this stage of our work, we were not able to
provide a meaningful explanation for this effect.
The time-adaptive algorithm with the saturation-based time-stepping scheme is
the most efficient for an L2-norm greater than 10-4. For more
accurate results, the iterative method with the time-stepping strategy using
the truncation error must be preferred. The impact of the time-stepping
strategy for these two algorithms is shown in Fig. 9 for approximately the
same L2-norm (2.051 × 10-3 within 1283 iterations for
TA_S and 1.517 × 10-3 within 6504 iterations for ST_Δψ). The time-step changes are related to the boundary condition
variations, as expected. The strategy based on the saturation variation leads
to a longer time step than the strategy using the time truncation error. This
difference can be quite important (see the simulation between days 25
and 30). The consequences of this difference are a reduced number of
iterations but also a less accurate computation, irrespective of the error
norm.
Summary and conclusions
The solution of RE is complex and very time-consuming due to its highly
nonlinear properties. Several algorithms have been tested for the mixed form
of the Richards equation, including time-adaptive methods. Based on the
numerical examples that differ in their parameters (level of nonlinearity)
and in their initial and boundary conditions, the conclusions and
recommendations are the following.
Our numerical developments showed that the method suggested by
Ross (2003) in its implicit formulation can be considered as a
Newton–Raphson method with a time-adaptive algorithm.
The different algorithms have different convergence rates (accuracy
improvement of the scheme as a function of the computational costs).
Therefore, an algorithm can be very efficient for a given accuracy and less
efficient for another level of precision. However, for these three test cases
and, on average, the best performance in terms of efficiency was obtained
using a stopping criterion based on truncation error with its corresponding
time-step strategy (ST_Δψ). Similar results were obtained by
Kavetski et al. (2001) for the pressure-based RE and by Kavetski and
Binning (2004) for the moisture-based RE.
The mass balance is not a good criterion for the evaluation of the results
because the mixed form preserves the mass balance, irrespective of the
pressure distribution within the profile.
The time truncation error should be implemented in numerical codes using the
standard iterative procedure. The use of the maximum variable difference
between two successive iterations only, which is usually implemented, does
not provide any information about the accuracy of the time derivative
approximation.
Our one-dimensional examples showed that the time-adaptive algorithm TA_T is
very sensitive to the type of problem to be solved. The time-adaptive
algorithm TA_S was less efficient than the usual schemes. However, for a
larger number of elements like in two-dimensional or
three-dimensional problems, this conclusion might be different because
the time dedicated to the computation of the parameters can be significantly
higher, unless tabulated values are used to evaluate the parameters and the
required derivatives.
Depending on the type of the problem that must be solved (parameter behavior
with respect to the pressure, time variations of the boundary conditions),
the time truncation errors may be predominant compared to the error
corresponding to the pressure changes between two successive iterations.
Therefore, we recommend the implementation of this stopping criteria
associated with the time-stepping strategy as defined by Kavetski et
al. (2001).
We did not use data. The computer program is available upon request.
The numerical method used in the paper is implicit standard finite
difference. For a cell i of the grid, the unsaturated flow Eq. (4) can
be discretized in the following way:
θin+1-θinΔt+Sws0ψin+1-ψinΔt+qi+n+1-qi-n+1Δzi=fi,qi-n+1=-Ki-ψin+1-ψi-1n+1Δzi--1,qi+n+1=-Ki+ψi+1n+1-ψin+1Δzi+-1,
where n is the time step, Ki- is the inter-block conductivity between
cell i and (i-1) defined by Ki-=Δzi-1K(ψi-1)+ΔziK(ψi)Δzi-1+Δzi, and Ki+ is
the inter-block conductivity between cell i and (i+1) defined by
Ki+=ΔziK(ψi)+Δzi+1K(ψi+1)Δzi+Δzi+1. Δzi-=12Δzi-1+Δzi is the distance between the center of cell
(i-1) and i. Δzi+=12Δzi+Δzi+1 is the distance between the center of cell i and
(i+1).
The residual is
R(ψin+1,k)=Δziθin+1,k-θin+ΔziSws0ψin+1,k-ψin+Δtqi+n+1,k-qi-n+1,k-ΔtΔzifi
where k is the iteration counter.
The residual derivatives are
∂R(ψin+1,k)∂ψi-1n+1,k=-Δt∂qi-n+1,k∂ψi-1n+1,k,∂R(ψin+1,k)∂ψin+1,k=Δzidθin+1,kdψin+1,k+ΔziSws0+Δt∂qi+n+1,k∂ψin+1,k-∂qi-n+1,k∂ψin+1,k,∂R(ψin+1,k)∂ψi+1n+1,k=Δt∂qi+n+1,k∂ψi+1n+1,k.
Therefore, the system to be solved is
-Δt∂qi-n+1,k∂ψi-1n+1,kΔψi-1n+1,k+1+Δzidθin+1,kdψin+1,k+ΔziSws0+Δt∂qi+n+1,k∂ψin+1,k-∂qi-n+1,k∂ψin+1,kΔψin+1,k+1+Δt∂qi+n+1∂ψi+1n+1,kΔψi+1n+1,k+1=-Δziθin+1,k-θin-ΔziSws0ψin+1,k-ψin-Δtqi+n+1,k-qi-n+1,k+ΔtΔzifi,
with the following derivatives of the fluxes qi-n+1,k,
∂qi-n+1,k∂ψi-1n+1,k=-∂Ki-n+1,k∂ψi-1n+1,kψin+1,k-ψi-1n+1,kΔzi--1+Ki-n+1,kΔzi-,∂qi-n+1,k∂ψin+1,k=-∂Ki-n+1,k∂ψin+1,kψin+1,k-ψi-1n+1,kΔzi--1-Ki-n+1,kΔzi-,
and qi+n+1,k:
∂qi+n+1,k∂ψin+1,k=-∂Ki+n+1,k∂ψin+1,kψi+1n+1,k-ψin+1,kΔzi+-1+Ki+n+1,kΔzi+,∂qi+n+1,k∂ψi+1n+1,k=-∂Ki+n+1,k∂ψi+1n+1,kψi+1n+1,k-ψin+1,kΔzi+-1-Ki+n+1,kΔzi+.
The component of the vector of the residuals R is given by Eq. (A2)
and the coefficients of the matrix R′ for cell i are
Ri-1,i′=Δt∂Ki-n+1,k∂ψi-1n+1,kψin+1,k-ψi-1n+1,kΔzi--1-Ki-n+1,kΔzi-,Ri,i′=Δzidθin+1,kdψin+1,k+Sws0-Δt∂Ki+n+1,k∂ψin+1,kψi+1n+1,k-ψin+1,kΔzi+-Ki+n+1,kΔzi++Δt∂Ki-n+1,k∂ψin+1,kψin+1,k-ψi-1n+1,kΔzi--1+Ki-n+1,kΔzi-,Ri,i+1′=-Δt∂Ki+n+1,k∂ψi+1n+1,kψi+1n+1,k-ψin+1,kΔzi+-1+Ki+n+1,kΔzi+.
In the case of prescribed flux at the upper boundary, the residual is written
as
R1(ψ1n+1,k)=Δz1θ1n+1,k-θ1n+Sws0ψ1n+1,k-ψ1n+Δtq1+n+1-qBC-ΔtΔz1f1.
Using the derivatives as defined in Eqs. (A5) and (A6), the matrix
coefficients are changed as follows:
R1,1′=Δz1dθ1n+1,kdψ1n+1,k+Sws0-Δt∂K1+n+1,k∂ψ1n+1,kψ2n+1,k-ψ1n+1,kΔz1+-1-K1+n+1,kΔz1+,R1,2′=-Δt∂K1+n+1,k∂ψ2n+1,kψ2n+1,k-ψ1n+1,kΔz1+-1+K1+n+1,kΔz1+.
If the flux is applied at the bottom of the profile, similar developments
lead to the residual
RN=ΔzNθNn+1,k-θNn+Sws0ψNn+1,k-ψNn+ΔtqBC-qN-n+1,k-ΔtΔzNfN
and its derivatives
RN-1,N′=Δt∂KN-n+1,k∂ψN-1n+1,kψNn+1,k-ψN-1n+1,kΔzN--1-KN-n+1,kΔzN-,RN,N′=ΔzNdθNn+1,kdψNn+1,k+Sws0+Δt∂KN-n+1,k∂ψNn+1,kψNn+1,k-ψN-1n+1,kΔzN--1+KN-n+1,kΔzN-.
If the pressure is described at the top of the soil, the corresponding flux
is defined by
q1-n+1,k=-K1-ψ1n+1,k-ψBCΔz1/2-1,
and the derivative is
∂q1-n+1,k∂ψ1n+1,k=-∂K1-n+1,k∂ψ1n+1,kψ1n+1,k-ψBCΔz1/2-1-K1-n+1,kΔz1/2.
The corresponding residual and the matrix coefficients are
R1=Δz1θ1n+1,k-θ1n+Sws0ψ1n+1,k-ψ1n+Δtq1+n+1,k-q1-n+1,k-ΔtΔz1f1
and
R1,1′=Δz1dθ1n+1,kdψ1n+1,k+Sws0-Δt∂K1+n+1,k∂ψ1n+1,kψ2n+1,k-ψ1n+1,kΔz1+-1-K1+n+1,kΔz1++Δt∂K1-n+1,k∂ψ1n+1,kψ1n+1,k-ψBCΔz1/2-1+K1-n+1,kΔz1/2R1,2′=-Δt∂K1+n+1,k∂ψ2n+1,kψ2n+1,k-ψ1n+1,kΔz1+-1+K1+n+1,kΔz1+.
Similarly, if the pressure is prescribed at the soil column's bottom, we have
RN=ΔzNθNn+1,k-θNn+Sws0ψNn+1,k-ψNn+ΔtqN+n+1,k-qN-n+1,k-ΔtΔzNfN
and
RN-1,N′=Δt∂KN-n+1,k∂ψN-1n+1,kψNn+1,k-ψN-1n+1,kΔzN--1-KN-n+1,kΔzN-,RN,N′=ΔzNdθNn+1,kdψNn+1,k+Sws0-Δt∂KN+n+1,k∂ψNn+1,kψBC-ψNn+1,kΔzN/2-1-ΔtKN+n+1,kΔzN/2+Δt∂KN-n+1,k∂ψNn+1,kψNn+1,k-ψN-1n+1,kΔzN--1+KN-n+1,kΔzN-.
The numerical code is written in FORTRAN 90 and is available upon request.
The authors declare that they have no conflict of interest.
Acknowledgements
The authors thank the anonymous referees for constructive review comments
which improved the quality of the document.
Edited by: B. Berkowitz
Reviewed by: two anonymous referees
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