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Register a new userA fully discrete variational scheme for solving nonlinear Fokker-Planck equations in higher space dimensions
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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 $d=2$.
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We propose a fully discrete variational scheme for nonlinear evolution equations with gradient flow structure on the space of finite Radon measures on an interval with respect to a generalized version of the Wasserstein distance with nonlinear mobility. Our scheme relies on a spatially discrete approximation of the semi-discrete (in time) minimizing movement scheme for gradient flows. Performing a finite-volume discretization of the continuity equation appearing in the definition of the distance, we obtain a finite-dimensional convex minimization problem usable as an iterative scheme. We prove that solutions to the spatially discrete minimization problem converge to solutions of the spatially continuous original minimizing movement scheme using the theory of $Gamma$-convergence, and hence obtain convergence to a weak solution of the evolution equation in the continuous-time limit if the minimizing movement scheme converges. We illustrate our result with numerical simulations for several second- and fourth-order equations.
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