A moving mesh finite difference method for non-monotone solutions of non-equilibrium equations in porous media

TL;DR

This paper introduces an adaptive moving mesh finite difference method for solving non-equilibrium equations in porous media, improving accuracy and mesh quality through dynamic mesh redistribution and flux analysis.

Contribution

It develops a novel adaptive moving mesh finite difference approach with directional control and smoothing for non-equilibrium porous media equations, enhancing numerical accuracy.

Findings

High mesh quality achieved

Accurate solutions demonstrated in 1D and 2D experiments

Effective flux schemes compared

Abstract

An adaptive moving mesh finite difference method is presented to solve two types of equations with dynamic capillary pressure term 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 time-dependent monitor function with directional control is applied to redistribute the mesh grid in every time step, and 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.