Entropy dissipation semi-discretization schemes for Fokker-Planck equations
For researchers studying Fokker-Planck equations and gradient flows, this provides a structure-preserving numerical method that maintains key physical properties.
The paper proposes a new semi-discretization scheme for nonlinear Fokker-Planck equations that preserves entropy dissipation and converges to a discrete Gibbs measure at an exponential rate, demonstrated on numerical examples.
We propose a new semi-discretization scheme to approximate nonlinear Fokker-Planck equations, by exploiting the gradient flow structures with respect to the 2-Wasserstein metric. We discretize the underlying state by a finite graph and define a discrete 2-Wasserstein metric. Based on such metric, we introduce a dynamical system, which is a gradient flow of the discrete free energy. We prove that the new scheme maintains dissipativity of the free energy and converges to a discrete Gibbs measure at exponential (dissipation) rate. We exhibit these properties on several numerical examples.