Convergence of a semi-explicit scheme for a one dimensional periodic nonlocal eikonal equation modeling dislocation dynamics

TL;DR

This paper introduces a periodic model for a nonlocal eikonal equation related to dislocation dynamics, proves its convergence to the original model as the period increases, and develops a semi-explicit numerical scheme with proven convergence and numerical validation.

Contribution

It presents a new periodic formulation of a nonlocal eikonal equation, along with a semi-explicit numerical scheme and rigorous convergence analysis.

Findings

The periodic model converges to the original as period tends to infinity.

The semi-explicit scheme is well-posed and satisfies a discrete gradient entropy inequality.

Numerical experiments validate the theoretical convergence results.

Abstract

In this paper, we derive a periodic model from a one dimensional nonlocal eikonal equation set on the full space modeling dislocation dynamics. Thanks to a gradient entropy estimate, we show that this periodic model converges toward the initial one when the period goes to infinity. Moreover, we design a semi-explicit numerical scheme for the periodic model that we introduce. We show the well-posedness of the scheme and a discrete gradient entropy inequality. We also prove the convergence of the scheme and we present some numerical experiments.