An entropy satisfying discontinuous Galerkin method for nonlinear Fokker-Planck equations

TL;DR

This paper introduces a high-order discontinuous Galerkin method for nonlinear Fokker-Planck equations that preserves entropy dissipation, steady-states, and positivity, ensuring accurate long-time behavior in simulations.

Contribution

It develops a novel DG scheme that maintains entropy decay, steady-states, and positivity for nonlinear Fokker-Planck equations, with rigorous positivity conditions and efficient long-term accuracy.

Findings

Scheme satisfies discrete entropy dissipation law

Positivity preserved through reconstruction algorithm

Numerical examples confirm high-order accuracy and long-time efficiency

Abstract

We propose a high order discontinuous Galerkin (DG) method for solving nonlinear Fokker-Planck equations with a gradient flow structure. For some of these models it is known that the transient solutions converge to steady-states when time tends to infinity. The scheme is shown to satisfy a discrete version of the entropy dissipation law and preserve steady-states, therefore providing numerical solutions with satisfying long-time behavior. The positivity of numerical solutions is enforced through a reconstruction algorithm, based on positive cell averages. For the model with trivial potential, a parameter range sufficient for positivity preservation is rigorously established. For other cases, cell averages can be made positive at each time step by tuning the numerical flux parameters. A selected set of numerical examples is presented to confirm both the high-order accuracy and the efficiency to capture the large-time asymptotic.