96 research outputs found
A study on iterative methods for solving Richards` equation
This work concerns linearization methods for efficiently solving the
Richards` equation,a degenerate elliptic-parabolic equation which models flow
in saturated/unsaturated porous media.The discretization of Richards` equation
is based on backward Euler in time and Galerkin finite el-ements in space. The
most valuable linearization schemes for Richards` equation, i.e. the
Newtonmethod, the Picard method, the Picard/Newton method and theLscheme are
presented and theirperformance is comparatively studied. The convergence, the
computational time and the conditionnumbers for the underlying linear systems
are recorded. The convergence of theLscheme is theo-retically proved and the
convergence of the other methods is discussed. A new scheme is
proposed,theLscheme/Newton method which is more robust and quadratically
convergent. The linearizationmethods are tested on illustrative numerical
examples
A linear domain decomposition method for partially saturated flow in porous media
The Richards equation is a nonlinear parabolic equation that is commonly used
for modelling saturated/unsaturated flow in porous media. We assume that the
medium occupies a bounded Lipschitz domain partitioned into two disjoint
subdomains separated by a fixed interface . This leads to two problems
defined on the subdomains which are coupled through conditions expressing flux
and pressure continuity at . After an Euler implicit discretisation of
the resulting nonlinear subproblems a linear iterative (-type) domain
decomposition scheme is proposed. The convergence of the scheme is proved
rigorously. In the last part we present numerical results that are in line with
the theoretical finding, in particular the unconditional convergence of the
scheme. We further compare the scheme to other approaches not making use of a
domain decomposition. Namely, we compare to a Newton and a Picard scheme. We
show that the proposed scheme is more stable than the Newton scheme while
remaining comparable in computational time, even if no parallelisation is being
adopted. Finally we present a parametric study that can be used to optimize the
proposed scheme.Comment: 34 pages, 13 figures, 7 table
A linear domain decomposition method for two-phase flow in porous media
This article is a follow up of our submitted paper [11] in which a
decomposition of the Richards equation along two soil layers was discussed. A
decomposed problem was formulated and a decoupling and linearisation technique
was presented to solve the problem in each time step in a fixed point type
iteration. This article extends these ideas to the case of two-phase in porous
media and the convergence of the proposed domain decomposition method is
rigorously shown.Comment: 8 page
Iterative Linearisation Schemes for Doubly Degenerate Parabolic Equations
Mathematical models for flow and reactive transport in porous media often involve non-linear, degenerate parabolic equations. Their solutions have low regularity, and therefore lower order schemes are used for the numerical approximation. Here the backward Euler method is combined with a mixed finite element method, which results in a stable and locally mass-conservative scheme. At each time step one has to solve a non-linear algebraic system, for which one needs adequate iterative solvers. Finding robust ones is particularly challenging here, since the problems considered are double degenerate (i.e. two type of degeneracies are allowed: parabolic-elliptic and parabolic-hyperbolic). Commonly used schemes, like Newton and Picard, are defined either for non-degenerate problems, or after regularising the problem in the case of degenerate ones. Convergence is guaranteed only if the initial guess is sufficiently close to the solution, which translates into severe restrictions on the time step. Here we discuss an iterative linearisation scheme which builds on the L-scheme, and does not employ any regularisation. We prove its rigorous convergence, which is obtained for Hölder type non-linearities. Finally, we present numerical results confirming the theoretical ones, and compare the behaviour of the proposed scheme with schemes based on a regularisation step.acceptedVersio
Efficient Solvers for Nonstandard Models for Flow and Transport in Unsaturated Porous Media
We study several iterative methods for fully coupled flow and reactive transport in porous media. The resulting mathematical model is a coupled, nonlinear evolution system. The flow model component builds on the Richards equation, modified to incorporate nonstandard effects like dynamic capillarity and hysteresis, and a reactive transport equation for the solute. The two model components are strongly coupled. On one hand, the flow affects the concentration of the solute; on the other hand, the surface tension is a function of the solute, which impacts the capillary pressure and, consequently, the flow. After applying an Euler implicit scheme, we consider a set of iterative linearization schemes to solve the resulting nonlinear equations, including both monolithic and two splitting strategies. The latter include a canonical nonlinear splitting and an alternate linearized splitting, which appears to be overall faster in terms of numbers of iterations, based on our numerical studies. The (time discrete) system being nonlinear, we investigate different linearization methods. We consider the linearly convergent L-scheme, which converges unconditionally, and the Newton method, converging quadratically but subject to restrictions on the initial guess. Whenever hysteresis effects are included, the Newton method fails to converge. The L-scheme converges; nevertheless, it may require many iterations. This aspect is improved by using the Anderson acceleration. A thorough comparison of the different solving strategies is presented in five numerical examples, implemented in MRST, a toolbox based on MATLAB.publishedVersio
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