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

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.