Entropy Dissipation Semi-Discretization Schemes for Fokker-Planck Equations

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 in the space of probability densities. We discretize the underlying state by a finite graph and define a discrete 2-Wasserstein metric in the discrete probability space. Based on such metric, we introduce a gradient flow of the discrete free energy as semi discretization scheme. We prove that the 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.