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A Moving Mesh Finite Difference Method for Non-Monotone Solutions of Non-Equilibrium Equations in Porous Media

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Numerical methods Representations of solutions Flows in porous media; filtration; seepage Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  28 July 2017

Hong Zhang  and
Paul Andries Zegeling
Hong Zhang*
Affiliation:
Department of Mathematics, Utrecht University, P.O.Box 80.010, 3508TA Utrecht, The Netherlands
Paul Andries Zegeling*
Affiliation:
Department of Mathematics, Utrecht University, P.O.Box 80.010, 3508TA Utrecht, The Netherlands
*
*Corresponding author. Email addresses:H.Zhang4@uu.nl (H. Zhang), P.A.Zegeling@uu.nl (P. A. Zegeling)
*Corresponding author. Email addresses:H.Zhang4@uu.nl (H. Zhang), P.A.Zegeling@uu.nl (P. A. Zegeling)
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Abstract

An adaptive moving mesh finite difference method is presented to solve two types of equations with dynamic capillary pressure effect in porous media. One is the non-equilibrium Richards Equation and the other is the modified Buckley-Leverett equation. The governing equations are discretized with an adaptive moving mesh finite difference method in the space direction and an implicit-explicit method in the time direction. In order to obtain high quality meshes, an adaptive monitor function with directional control is applied to redistribute the mesh grid in every time step, then a diffusive mechanism is used to smooth the monitor function. The behaviors of the central difference flux, the standard local Lax-Friedrich flux and the local Lax-Friedrich flux with reconstruction are investigated by solving a 1D modified Buckley-Leverett equation. With the moving mesh technique, good mesh quality and high numerical accuracy are obtained. A collection of one-dimensional and two-dimensional numerical experiments is presented to demonstrate the accuracy and effectiveness of the proposed method.

Keywords

Relaxation non-equilibrium Richards equationmodified Buckley-Leverett equationsaturation overshoottraveling wave analysismoving mesh finite difference method

MSC classification

Secondary: 35C07: Traveling wave solutions 35Q35: PDEs in connection with fluid mechanics 65M50: Mesh generation and refinement 74S20: Finite difference methods 76S05: Flows in porous media; filtration; seepage
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 4 , October 2017 , pp. 935 - 964
Copyright
Copyright © Global-Science Press 2017 

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Footnotes

Communicated by Tao Zhou

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