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

Diana Al Zareef, Ahmad El Hajj, Antoine Zurek

TL;DR

This work addresses a two-dimensional nonlocal, non-conservative hyperbolic transport system for dislocation densities, characterized by weak regularity in the velocity and data. It introduces a Fejér-kernel regularization of the nonlocal kernel and an IMEX finite-difference scheme that preserves a discrete gradient-entropy inequality, enabling convergence to the regularized PDE and, subsequently, to the original model as the regularization is removed. The main contributions are (i) a rigorous scheme satisfying stability and entropy properties, (ii) a convergence proof for fixed regularization parameter $M$ and, crucially, (iii) aLimiting process as $M\to\infty$ to recover the non-regularized system, supported by compactness arguments (Simon, Orlicz spaces). Numerical experiments corroborate the theoretical findings, demonstrating the scheme's ability to capture the expected approach to a spatially uniform dislocation density under external and internal stress configurations. Overall, the paper provides a first convergence result for this 2D nonlocal, non-conservative hyperbolic system and delivers a robust computational tool for simulating mesoscale dislocation dynamics in periodic domains.

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 Fejér 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.

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

TL;DR

This work addresses a two-dimensional nonlocal, non-conservative hyperbolic transport system for dislocation densities, characterized by weak regularity in the velocity and data. It introduces a Fejér-kernel regularization of the nonlocal kernel and an IMEX finite-difference scheme that preserves a discrete gradient-entropy inequality, enabling convergence to the regularized PDE and, subsequently, to the original model as the regularization is removed. The main contributions are (i) a rigorous scheme satisfying stability and entropy properties, (ii) a convergence proof for fixed regularization parameter and, crucially, (iii) aLimiting process as to recover the non-regularized system, supported by compactness arguments (Simon, Orlicz spaces). Numerical experiments corroborate the theoretical findings, demonstrating the scheme's ability to capture the expected approach to a spatially uniform dislocation density under external and internal stress configurations. Overall, the paper provides a first convergence result for this 2D nonlocal, non-conservative hyperbolic system and delivers a robust computational tool for simulating mesoscale dislocation dynamics in periodic domains.

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 Fejér 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.
Paper Structure (29 sections, 18 theorems, 223 equations, 2 figures, 1 table)

This paper contains 29 sections, 18 theorems, 223 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Physical motivation
  3. Brief review of the literature
  4. Mathematical structure of the initial system and description of our results
  5. Derivation of a regularized model and gradient entropy estimate
  6. Outline of the paper
  7. Numerical scheme and main results
  8. Numerical scheme
  9. Main results
  10. Well-posedness and qualitative properties of the scheme
  11. Technical lemmas
  12. Well-posedness of the scheme
  13. Construction of a contraction mapping and first properties
  14. Nonnegativity of the discrete gradient and discrete $L^\infty$ estimate.
  15. Discrete gradient entropy estimate
  16. ...and 14 more sections

Key Result

Theorem 1

Let assumptions a and H2 hold, and assume that $N \geq M\geq 1$ with and Then, the scheme initial data--sigma_def admits a unique solution $\rho^{\pm,n,\mathrm{per}}_M = (\rho^{\pm,n,\mathrm{per}}_{M,i,j})_{(i,j)\in \mathcal{I}_N^2}$ for any $1\le n \le N_T$, such that and Moreover, for $0\le n \le N_T-1$, we have where we recall definition function f of $f$ and with $\mathcal{D}\left[\rho^{n

Figures (2)

  • Figure 1: Evolution of $\partial_{x_1}\rho^{+}$ under external stress.
  • Figure 2: Evolution of $\partial_{x_1}\rho^{\pm}$ under internal and external stresses.

Theorems & Definitions (33)

  • Remark 1
  • Remark 2
  • Theorem 1: Well-posedness of the scheme
  • Theorem 2: Convergence of the scheme
  • Theorem 3: Limit $M$ to $\infty$
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 23 more