Convergence of a scheme for a two dimensional nonlocal system of transport equations

TL;DR

This paper introduces a semi-explicit finite difference scheme for a two-dimensional nonlocal hyperbolic system modeling dislocation densities, proving its convergence despite weak regularity and non-strict hyperbolicity.

Contribution

The paper presents the first convergence proof for a numerical scheme applied to this class of nonlocal, weakly regular hyperbolic systems with coupled transport equations.

Findings

The scheme preserves a discrete entropy estimate.

Convergence of the scheme to the continuous solution is established.

Numerical illustrations demonstrate the scheme's effectiveness.

Abstract

In this paper, we numerically study a two-dimensional system modeling the dynamics of dislocation densities. This system is hyperbolic, but not strictly hyperbolic, and couples two non-local transport equations. It is characterized by weak regularity in both the velocity and the initial data. We propose a semi-explicit finite difference (IMEX) numerical scheme for the discretization of this system, after regularizing the singular velocity using a Fej\'er kernel. We show that this scheme preserves, at the discrete level, an entropy estimate on the gradient, which then allows us to establish the convergence of the discrete solution to the continuous solution. To our knowledge, this is the first convergence result obtained for this type of system. We conclude with some numerical illustrations highlighting the performance of the proposed scheme.