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An adaptive moving mesh finite difference method is presented to solve twotypes of equations with dynamic capillary pressure term in porous media. One isthe non-equilibrium Richards Equation and the other is the modifiedBuckley-Leverett equation. The governing equations are discretized with anadaptive moving mesh finite difference method in the space direction and animplicit-explicit method in the time direction. In order to obtain high qualitymeshes, an adaptive time-dependent monitor function with directional control isapplied to redistribute the mesh grid in every time step, and a diffusivemechanism is used to smooth the monitor function. The behaviors of the centraldifference flux, the standard local Lax-Friedrich flux and the localLax-Friedrich flux with reconstruction are investigated by solving a 1Dmodified Buckley-Leverett equation. With the moving mesh technique, good meshquality and high numerical accuracy are obtained. A collection ofone-dimensional and two-dimensional numerical experiments is presented todemonstrate the accuracy and effectiveness of the proposed method.

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