Diffusion front capturing schemes for a class of Fokker-Planck equations: application to the relativistic heat equation. (English) Zbl 1423.35365
Summary: 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.
MSC:
| 35Q84 | Fokker-Planck equations |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
diffusion fronts; Fokker-Planck equation; relativistic heat equation; conservative finite difference schemeReferences:
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