Discrete approximation of the minimizing movement scheme for evolution equations of Wasserstein gradient flow type with nonlinear mobility

TL;DR

This paper introduces a fully discrete variational scheme for nonlinear evolution equations with Wasserstein gradient flow structure, proving convergence of the discrete solutions to the continuous problem and demonstrating numerical effectiveness.

Contribution

It develops a spatially discrete approximation of the minimizing movement scheme for Wasserstein gradient flows with nonlinear mobility, including convergence analysis.

Findings

Convergence of the discrete scheme to the continuous solution via Γ-convergence.

Numerical simulations for second- and fourth-order equations.

A finite-dimensional convex minimization problem for iterative solutions.

Abstract

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.