Deterministic particle method for nonlinear nonlocal scalar balance equations

TL;DR

This paper introduces a deterministic particle scheme to solve nonlinear nonlocal scalar balance equations with source terms, addressing the challenge of non-probability measure solutions in congested dynamics modeling.

Contribution

It extends previous particle methods by incorporating source terms, adapting the scheme and estimates for solutions that are not probability measures.

Findings

Successfully handles source terms in particle schemes

Ensures compactness of solutions despite non-probability measures

Provides a robust numerical approach for congested dynamics modeling

Abstract

We study a deterministic particle scheme to solve a scalar balance equation with nonlocal interaction and nonlinear mobility used to model congested dynamics. The main novelty with respect to "Radici-Stra [SIAM J. Math. Anal. 55.3 (2023)]" is the presence of a source term; this causes the solutions to no longer be probability measures, thus requiring a suitable adaptation of the numerical scheme and of the estimates leading to compactness.