The following work have been accepted for publication on "Journal of Computational and Applied Mathematics" https://doi.org/10.1016/j.cam.2024.116325:

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

Eduardo Abreu, Richard A. De la Cruz Guerrero, Juan Carlos Juajibioy Otero and Wanderson José Lambert.

Highlights:
In this work, we have expanded upon the (local) semi-discrete Lagrangian-Eulerian method initially introduced in [E. Abreu, J. Francois, W. Lambert and J. Perez (2022), https://doi.org/10.1016/j.cam.2021.114011] to approximate a specific class of multi-dimensional scalar conservation laws with nonlocal flux. For completeness, we analyze the convergence of this method using the weak asymptotic approach introduced in [Eduardo Abreu, Mathilde Colombeau and Evgeny Yu Panov (2016), https://doi.org/10.1016/j.jmaa.2016.06.047], 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 L1(Rd) ∩ L∞(Rd) and 2) The family is dominated by a suitable temporal and spatial modulus of continuity. These properties allow us to employ the L1-compactness argument to extract a convergent subsequence. We demonstrate that the limit function is a weak entropy solution. Finally, we present a section of numerical examples in 1D and also in 2D for two-dimensional nonlocal Burgers equations to illustrate our results.