Diffusion front capturing schemes for a class of Fokker-Planck equations: Application to the relativistic heat equation
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-Friedrich-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 (C) 2009 Elsevier Inc. All rights reserved.
diffusion fronts fokker-planck equation relativistic heat equation conservative finite difference scheme