A Robust and Efficient Linearization Scheme for Doubly Nonlinear and Degenerate Parabolic Problems Arising in Flow in Porous Media

Slodička, Marián

doi:10.1137/s1064827500381860

Cited by 50 publications

(62 citation statements)

References 28 publications

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“…We note that the scheme presented above is slightly different from the one considered in [32], where…”

Section: Remarkmentioning

confidence: 95%

“…The method still involves the computation of derivatives and in the degenerate case might also fail to converge (see the numerical examples in Section 4). The L−method was proposed for Richards' equation by [25,32,36] and it is the only method which exploits the monotonicity of θ(·). The L−scheme to solve the non-linear problem (4) reads:…”

Section: Linearization Methods For Richards' Equationmentioning

confidence: 99%

“…As regards the spatial discretization there are much more options possible. Galerkin finite elements were used in [2,3,9,22,24,32], often together with mass lumping to ensure a maximum principle [8]. Locally mass conservative schemes for Richards' equation were proposed and analysed in [10,11] (finite volumes), in [16] (multipoint flux approximation) or [4,5,26,29,34,35] (mixed finite element method).…”

Section: ∂ T θ(ψ ) − ∇ · (K(θ(ψ ))∇(ψmentioning

confidence: 99%

“…It was proposed for Richards' equation in [25,32,36]. The method is robust and linearly convergent, and does not involve the computation of any derivatives.…”

Section: ∂ T θ(ψ ) − ∇ · (K(θ(ψ ))∇(ψmentioning

confidence: 99%

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A study on iterative methods for solving Richards’ equation

2016

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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 elements in space. The most valuable linearization schemes for Richards' equation, i.e. the Newton method, the Picard method, the Picard/Newton method and the L−scheme are presented and their performance is comparatively studied. The convergence, the computational time and the condition numbers for the underlying linear systems are recorded. The convergence of the L−scheme is theoretically proved and the convergence of the other methods is discussed. A new scheme is proposed, the L−scheme/Newton method which is more robust and quadratically convergent. The linearization methods are tested on illustrative numerical examples.

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“…We note that the scheme presented above is slightly different from the one considered in [32], where…”

Section: Remarkmentioning

confidence: 95%

Section: Linearization Methods For Richards' Equationmentioning

confidence: 99%

Section: ∂ T θ(ψ ) − ∇ · (K(θ(ψ ))∇(ψmentioning

confidence: 99%

“…It was proposed for Richards' equation in [25,32,36]. The method is robust and linearly convergent, and does not involve the computation of any derivatives.…”

Section: ∂ T θ(ψ ) − ∇ · (K(θ(ψ ))∇(ψmentioning

confidence: 99%

See 2 more Smart Citations

A study on iterative methods for solving Richards’ equation

2016

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“…In [20], this approach is placed in a fixed point context, for approximating the solution of an elliptic problem with a nonlinear and possibly unbounded source term (see also [27]). The same ideas are followed in [23] and [19], where similar schemes are considered for the implicit discretization of a degenerate (fast diffusion) problem in both conformal and mixed formulation. We also mention here [24] for a related work on nonlinear elliptic equations.…”

Section: A Fixed Point Iteration For the Time Discrete Problemsmentioning

confidence: 99%

A Numerical Scheme for the Pore-Scale Simulation of Crystal Dissolution and Precipitation in Porous Media

Devigne¹,

Pop²,

Duijn³

et al. 2008

SIAM J. Numer. Anal.

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Abstract.In this paper we analyze a numerical scheme for a dissolution and precipitation in porous media. We focus here on the chemistry, which is modeled by a parabolic problem that is coupled through the boundary conditions to an ordinary differential inclusion defined on the boundary. We use a regularization approach for constructing a semi-implicit scheme that is stable and convergent. For dealing with the emerging time discrete nonlinear problems, we propose a simple fixed point iterative procedure. The paper is concluded by numerical results.

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An improved numerical simulator for multiphase flow in porous media

Garrido

Fladmark

Espedal

2004

Numerical Methods in Fluids

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SUMMARYIn basin modelling the thermodynamics of a multicomponent multiphase uid ux are computationally too expensive when derived from an equation of state and the Gibbs equality constraints. In this article we present a novel implicit molar mass formulation technique using binary mixture thermodynamics. The two proposed solution methods, with and without cross derivative terms between components, are based on a preconditioned Newton-GMRES scheme for each time-step with analytical computation of the derivatives. These new algorithms reduce signiÿcantly the numerical e ort for the computation of the molar masses, and we illustrate the behavior of these methods with numerical computations.

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A Robust and Efficient Linearization Scheme for Doubly Nonlinear and Degenerate Parabolic Problems Arising in Flow in Porous Media

Cited by 50 publications

References 28 publications

A study on iterative methods for solving Richards’ equation

A study on iterative methods for solving Richards’ equation

A Numerical Scheme for the Pore-Scale Simulation of Crystal Dissolution and Precipitation in Porous Media

An improved numerical simulator for multiphase flow in porous media

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