| Issue |
ESAIM: M2AN
Volume 59, Number 6, November-December 2025
|
|
|---|---|---|
| Page(s) | 2957 - 2990 | |
| DOI | https://doi.org/10.1051/m2an/2025080 | |
| Published online | 07 November 2025 | |
Convergence of a semi-explicit scheme for a one dimensional periodic nonlocal eikonal equation modeling dislocation dynamics
1
Université de Technologie de Compiègne, LMAC, 60200 Compiègne, France
2
Faculty of Science (I), Mathematics Department, Lebanese University, Hadath, Lebanon
3
CNRS – Université de Montréal CRM – CNRS, Montreal, Canada
* Corresponding author: antoine.zurek@utc.fr
Received:
29
January
2025
Accepted:
18
September
2025
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.
Mathematics Subject Classification: 74S20 / 65M12 / 35F20 / 35Q74
Key words: Nonlocal eikonal equation / semi-explicit upwind scheme / gradient entropy estimate / dislocation dynamics
Publisher note: The first author's second email address has been added. This article was corrected on November 20, 2025.
© The authors. Published by EDP Sciences, SMAI 2025
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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