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

摘要

We propose a high order discontinuous Galerkin (DG) method for solvingnonlinear Fokker-Planck equations with a gradient flow structure. For some ofthese models it is known that the transient solutions converge to steady-stateswhen time tends to infinity. The scheme is shown to satisfy a discrete versionof the entropy dissipation law and preserve steady-states, therefore providingnumerical solutions with satisfying long-time behavior. The positivity ofnumerical solutions is enforced through a reconstruction algorithm, based onpositive cell averages. For the model with trivial potential, a parameter rangesufficient for positivity preservation is rigorously established. For othercases, cell averages can be made positive at each time step by tuning thenumerical flux parameters. A selected set of numerical examples is presented toconfirm both the high-order accuracy and the efficiency to capture thelarge-time asymptotic.