Stabilized Finite Difference Methods for the Fully Dynamic Biot's Problem

Abstract

This paper deals with the stabilization of the poroelasticity system, in the incompressible fully dynamic case. The stabilization term is a perturbation of the equilibrium equation that allows us to use central difference schemes to approximate the first order spatial derivatives, yielding numerical solutions without oscillations independently of the chosen discretization parameters. The perturbation term is a discrete Laplacian of the forward time difference, affected by a stabilization parameter depending on the mesh size and the properties of the porous medium. In the one dimensional case, this parameter is shown to be optimal. Some numerical experiments are presented to show the efficiency of the proposed stabilization technique.

Keywords:

• poroelasticity
• stabilization
• finite-differences
• three-level schemes

AMS Subject Classification:

• 65N06
• 65M06
• 74F10
• 74S20