Diffusion front capturing schemes for a class of Fokker-Planck equations: Application to the relativistic heat equation

Authors:
Antonio Marquina
Affiliations:
Departmento de Matematica Aplicada, Universidad de Valencia, Avda. Dr. Moliner 50, Burjassot-Valencia 46100, Spain
Venue:
Journal of Computational Physics
Year:
2010

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Abstract

In this research work we introduce and analyze an explicit conservative finite difference scheme to approximate the solution of initial-boundary value problems for a class of limited diffusion Fokker-Planck equations under homogeneous Neumann boundary conditions. We show stability and positivity preserving property under a Courant-Friedrichs-Lewy parabolic time step restriction. We focus on the relativistic heat equation as a model problem of the mentioned limited diffusion Fokker-Planck equations. We analyze its dynamics and observe the presence of a singular flux and an implicit combination of nonlinear effects that include anisotropic diffusion and hyperbolic transport. We present numerical approximations of the solution of the relativistic heat equation for a set of examples in one and two dimensions including continuous initial data that develops jump discontinuities in finite time. We perform the numerical experiments through a class of explicit high order accurate conservative and stable numerical schemes and a semi-implicit nonlinear Crank-Nicolson type scheme.