Semi-discrete Lagrangian-Eulerian approach based on the weak asymptotic method for nonlocal conservation laws in several dimensions

Abstract

In this work, we have expanded upon the (local) semi-discrete Lagrangian-Eulerian method initially introduced in Abreu et al. (2022) to approximate a specific class of multi-dimensional scalar conservation laws with nonlocal flux, referred to as the nonlocal model: ${\partial }_t\rho (t,x)+{\sum }_{i=1}^d{\partial }_{x_i}\left(V^i\left[W[\rho ,\omega ](t,x)\right]F^i\left(\rho (t,x)\right)\right)=0,(t,x)\in (0,T)\times R^d.$ For completeness, we analyze the convergence of this method using the weak asymptotic approach introduced in Abreu et al. (2016), with significant results extended to the multidimensional nonlocal case. While there are indeed other important techniques available that can be utilized to prove the convergence of the numerical scheme, the choice of this particular technique (weak asymptotic analysis) is quite natural. This is primarily due to its suitability for dealing with the Lagrangian-Eulerian schemes proposed in this paper. Essentially, the weak asymptotic method generates a family of approximate solutions satisfying the following properties: 1) The family of approximate functions is uniformly bounded in the space $L^1\left(R^d\right)\cap L^{\infty }\left(R^d\right)$. 2) The family is dominated by a suitable temporal and spatial modulus of continuity. These properties allow us to employ the $L^1$-compactness argument to extract a convergent subsequence. We demonstrate that the limit function is a weak entropy solution of Eq. (1). Finally, we present a section of numerical examples to illustrate our results. Finally, we have examined examples discussed in Aggarwal et al. (2015) and Keimer et al. (2018). In the context of the family of two-dimensional nonlocal Burgers equations, we provide numerical results for a nonlocal impact of the form ${\omega }_{\eta }\ast \rho$, where $\eta =0.1$.