A fully discrete variational scheme for solving nonlinear Fokker-Planck equations in higher space dimensions
This work provides a novel numerical method for solving nonlinear Fokker-Planck equations in higher dimensions, which is important for applications in physics and biology, but the results are limited to 2D experiments and theoretical convergence under assumptions.
The authors propose a new spatio-temporal discretization for nonlinear Fokker-Planck equations on multi-dimensional unit cubes, which is entropy diminishing, mass conserving, and weakly stable, with convergence proven under regularity assumptions and demonstrated in 2D numerical experiments.
We introduce a novel spatio-temporal discretization for nonlinear Fokker-Planck equations on the multi-dimensional unit cube. This discretization is based on two structural properties of these equations: the first is the representation as a gradient flow of an entropy functional in the $L^2$-Wasserstein metric, the second is the Lagrangian nature, meaning that solutions can be written as the push forward transformation of the initial density under suitable flow maps. The resulting numerical scheme is entropy diminishing and mass conserving. Further, the scheme is weakly stable, which allows us to prove convergence under certain regularity assumptions. Finally, we present results from numerical experiments in space dimension $d=2$.