The Entropy Satisfying Discontinuous Galerkin Method for Fokker-Planck equations

In Liu and Yu (SIAM J Numer Anal 50(3):1207–1239, 2012), we developed a finite volume method for Fokker–Planck equations with an application to finitely extensible nonlinear elastic dumbbell model for polymers subject to homogeneous fluids. The method preserves positivity and satisfies the discrete entropy inequalities, but has only first order accuracy in general cases. In this paper, we overcome this problem by developing uniformly accurate, entropy satisfying discontinuous Galerkin methods for solving Fokker– Planck equations. Both semidiscrete and fully discretemethods satisfy two desired properties: mass conservation and entropy satisfying in the sense that these schemes are shown to satisfy the discrete entropy inequality. These ensure that the schemes are entropy satisfying and preserve the equilibrium solutions. It is also proved the convergence of numerical solutions to the equilibrium solution as time becomes large. At the finite time, a positive truncation is used to generate the nonnegative numerical approximation which is as accurate as the obtained numerical solution. Both one and two-dimensional numerical results are provided to demonstrate the good qualities of the schemes, as well as effects of some canonical homogeneous flows.