A finite volume scheme preserving the invariant region property for the coupled system of FitzHugh-Nagumo equations on distorted meshes

Abstract

In this paper, we propose and analyze a finite volume scheme preserving the invariant region property (IRP) for the coupled system of equations of FitzHugh-Nagumo type on distorted meshes. We employ a nonlinear cell-centered finite volume scheme to approximate the flux and implicit Euler scheme to approximate the time derivative. The IRP and existence of solutions are proved for the nonlinear finite volume scheme. We also propose a nonlinear iteration method which adds a specifically designed penalization term to ensure the IRP on each iteration step. Numerical experiments and the comparison with the nine-point finite volume scheme are presented to verify the IRP of our scheme.