Discretization and convergence of the ballistic Benamou-Brenier formulation of the porous medium and Burgers equations

TL;DR

This paper develops a stable, convergent discretization method for the porous medium and Burgers equations using a reformulation of optimal transport, enabling efficient numerical solutions and demonstrating quadratic convergence.

Contribution

It introduces a novel discretization scheme with harmonic interpolation for these equations, proving stability and convergence, and implements an efficient numerical solver.

Findings

Unconditionally stable schemes with respect to space and time steps.

Quadratic convergence rate for the dual PDE solution.

Efficient numerical implementation using proximal splitting and Fourier transform.

Abstract

We study the discretization, convergence, and numerical implementation of recent reformulations of the quadratic porous medium equation (multidimensional and anisotropic) and Burgers' equation (one-dimensional, with optional viscosity), as forward in time variants of the Benamou-Brenier formulation of optimal transport. This approach turns those evolution problems into global optimization problems in time and space, of which we introduce a discretization, one of whose originalities lies in the harmonic interpolation of the densities involved. We prove that the resulting schemes are unconditionally stable w.r.t. the space and time steps, and we establish a quadratic convergence rate for the dual PDE solution, under suitable assumptions. We also show that the schemes can be efficiently solved numerically using a proximal splitting method and a global space-time fast Fourier transform, and we illustrate our results with numerical experiments.