Monotone-based Numerical Schemes for Two-Dimensional Systems of Nonlocal Conservation Laws

TL;DR

This paper introduces a class of monotone numerical schemes for two-dimensional nonlocal conservation laws, providing convergence guarantees, error estimates, and numerical validation for a broad range of models.

Contribution

The paper develops a general framework for monotone schemes applied to 2D nonlocal systems, including convergence proofs, error bounds, and existence and uniqueness results.

Findings

Schemes converge to the unique weak entropy solution.

Error estimate of order √Δt for the numerical approximation.

Numerical experiments confirm theoretical convergence and accuracy.

Abstract

We present a class of numerical schemes for two-dimensional systems of nonlocal conservation laws, which are based on utilizing well-known monotone numerical flux functions after suitably approximating the nonlocal terms. The considered systems are weakly coupled by the nonlocal terms and the underlying flux function is rather general to guarantee that our results are applicable to a wide range of common nonlocal models. We state sufficient conditions to ensure the convergence of the monotone-based numerical schemes to the unique weak entropy solution. Moreover, we provide an error estimate that yields the convergence rate of $\mathcal{O}(\sqrt{\Delta t})$ for the numerical approximations of the solution. Our results include an existence and uniqueness proof of the nonlocal system, too. Numerical results illustrate our theoretical findings.