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

¹ University Paris-Saclay, ENS Paris-Saclay, CNRS, Centre Borelli, F-91190 Gif-sur-Yvette, France
² University Paris-Saclay, Laboratoires de Mathématiques d’Orsay, 91400 Orsay, France

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 6 April 2025
Accepted: 15 December 2025

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.