A discontinuous Galerkin method for the mono-energetic Fokker–Planck equation based on a spherical interior penalty formulation

Abstract

We present a new discretization of the mono-energetic Fokker–Planck equation. We build on previous work (Kópházi and Lathouwers, 2015) where we devised an angular discretization for the Boltzmann equation, allowing for both heterogeneous and anisotropic angular refinement. The angular discretization is based on a discontinuous finite element method on the unit sphere. Here we extend the methodology to include the effect of the Fokker–Planck scatter operator describing small angle particle scatter. We describe the construction of an interior penalty method on the sphere surface. Results are provided for a variety of test cases, ranging from purely angular to fully three-dimensional. The results show that the scheme can resolve highly forward-peaked flux distributions with forward-peaked scatter.