A convergent Lagrangian discretization for $p$-Wasserstein and flux-limited diffusion equations

TL;DR

This paper introduces a Lagrangian discretization scheme for nonlinear drift diffusion equations based on optimal transport, demonstrating convergence and preserving key properties like entropy monotonicity and flux-limitation.

Contribution

The paper develops a novel Lagrangian numerical scheme for $p$-Wasserstein and flux-limited diffusion equations, with proven convergence and property preservation.

Findings

Scheme inherits entropy monotonicity and mass preservation.

Convergence proven as mesh size vanishes.

Numerical experiments confirm theoretical results.

Abstract

We study a Lagrangian numerical scheme for solution of a nonlinear drift diffusion equation of the form $\partial_t u = \partial_x(u \cdot c[\partial_x(h^\prime(u)+v)])$ on an interval. This scheme will consist of a spatio-temporal discretization founded in the formulation of the equation in terms of inverse distribution functions. It is based on the gradient flow structure of the equation with respect to optimal transport distances for a family of costs that are in some sense $p$-Wasserstein like. Additionally we will show that, under a regularity assumption on the initial data, this also includes a family of discontinuous, flux-limiting cost. We show that this discretization inherits various properties from the continuous flow, like entropy monotonicity, mass preservation, a minimum/maximum principle and flux-limitation in the case of the corresponding cost. Convergence in the limit of vanishing mesh size will be proven as the main result. Finally we will present numerical experiments including a numerical convergence analysis.