Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms

TL;DR

This paper introduces a numerical scheme for nonlinear continuity equations based on gradient flow and Wasserstein distance, ensuring positivity and energy decrease, applicable to various internal energies and external potentials.

Contribution

It presents a new variational numerical method for nonlinear continuity equations using evolving diffeomorphisms and Wasserstein gradient flows.

Findings

Scheme guarantees positivity of solutions.

Energy decreases in the semi-discrete scheme.

Effective in 1D and 2D examples.

Abstract

In this paper we present a numerical scheme for nonlinear continuity equations, which is based on the gradient flow formulation of an energy functional with respect to the quadratic transportation distance. It can be applied to a large class of nonlinear continuity equations, whose dynamics are driven by internal energies, given external potentials and/or interaction energies. The solver is based on its variational formulation as a gradient flow with respect to the Wasserstein distance. Positivity of solutions as well as energy decrease of the semi-discrete scheme are guaranteed by its construction. We illustrate this properties with various examples in spatial dimension one and two.