Discretization of Flux-Limited Gradient Flows: $\Gamma$-convergence and numerical schemes

TL;DR

This paper develops a numerical scheme for discretizing flux-limited gradient flows, including the relativistic heat equation, ensuring entropy dissipation, mass conservation, and finite propagation speed, with proven convergence properties.

Contribution

It introduces a novel discretization combining the JKO method with entropic regularization and Dykstra's algorithm, along with a proof of $ extGamma$-convergence for the scheme.

Findings

The scheme preserves key physical properties like entropy dissipation and finite speed of propagation.

The $ extGamma$-convergence proof handles the singular cost function in the discretization.

Numerical experiments demonstrate the efficiency and accuracy of the proposed method.

Abstract

We study a discretization in space and time for a class of nonlinear diffusion equations with flux limitation. That class contains the so-called relativistic heat equation, as well as other gradient flows of Renyi entropies with respect to transportation metrics with finite maximal velocity. Discretization in time is performed with the JKO method, thus preserving the variational structure of the gradient flow. This is combined with an entropic regularization of the transport distance, which allows for an efficient numerical calculation of the JKO minimizers. Solutions to the fully discrete equations are entropy dissipating, mass conserving, and respect the finite speed of propagation of support. First, we give a proof of $\Gamma$-convergence of the infinite chain of JKO steps in the joint limit of infinitely refined spatial discretization and vanishing entropic regularization. The singularity of the cost function makes the construction of the recovery sequence significantly more difficult than in the $L^p$-Wasserstein case. Second, we define a practical numerical method by combining the JKO time discretization with a "light speed" solver for the spatially discrete minimization problem using Dykstra's algorithm, and demonstrate its efficiency in a series of experiments.