Positive Second Order Finite Difference Methods on Fokker-Planck Equations with Dirac Initial Data

Abstract

The backward Euler scheme is known to be monotone and positivity preserving. Those are particularly important properties when the partial differential equation represents a density with a discontinuous initial condition like a Dirac. Many schemes will produce oscillations or negative densities. This paper analyzes the behavior of a few simple schemes related to backward Euler, namely BDF2, and Lawson-Morris on the specific problem of a diffusion with Dirac initial condition.