  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.
