Maximum-Principle-Satisfying Third Order Discontinuous Galerkin Schemes for Fokker-Planck Equations

We design and analyze up to third order accurate discontinuous Galerkin (DG) methods satisfying a strict maximum principle for Fokker–Planck equations. A procedure is established to identify an effective test set in each computational cell to ensure the desired bounds of numerical averages during time evolution. This is achievable by taking advantage of the two parameters in the numerical flux and a novel decomposition of weighted cell averages. Based on this result, a scaling limiter for the DG method with first order Euler forward time discretization is proposed to solve the one-dimensional Fokker–Planck equations. Strong stability preserving high order time discretizations will keep the maximum principle. It is straightforward to extend the method to two and higher dimensions on rectangular meshes. We also show that a modified limiter can preserve the strict maximum principle for DG schemes solving Fokker–Planck equations. As a consequence, the present schemes preserve steady states. Numerical tests for the DG method are reported, with applications to polymer models with both Hookean and FENE potentials.