An Efficient Positivity-Preserving Finite Difference Scheme for Solving the Fokker-Planck Diffusion Equation

The Fokker-Planck diffusion equation is widely used for simulating the evolution of Earth's radiation belt electrons, which pose significant hazards to space-borne systems. To preserve the positivity of the numerical solution of the electron phase space density (PSD), several finely designed finite difference, Monte Carlo, spatiotemporal interpolation, and finite volume schemes have been developed. However, these schemes often suffer from either high implementation complexity or low execution efficiency. Here we propose an efficient, easy-to-implement, and positivity-preserving finite difference scheme, named the Semi-Implicit Logarithmic Linearization (SILL) scheme. The basic principle is to linearize the nonlinear equation of the natural logarithmic PSD. This scheme ensures accuracy and stability, even with large time steps, up to hundreds of seconds for typical radiation belt electron diffusion processes. Nonetheless, it exhibits oversensitivity to near-vanishing phase space densities, which necessitates reduced time steps when handling extremely large variations in orders of magnitude between neighboring grid points. We have publicly released the protype code of the SILL scheme, which could be useful for the radiation belt modeling community.