# On the role of numerical viscosity in the study of the local limit of nonlocal conservation laws

## TL;DR

This paper investigates how numerical viscosity affects the study of the local limit of nonlocal conservation laws, revealing that certain schemes may falsely indicate convergence due to numerical artifacts.

## Contribution

It demonstrates that Lax-Friedrichs schemes' numerical viscosity can mislead convergence analysis, and shows that Godunov schemes offer more reliable results in this context.

## Key findings

- Lax-Friedrichs schemes can falsely suggest convergence due to numerical viscosity.
- Godunov schemes are less affected by numerical viscosity and provide more accurate insights.
- Numerical viscosity plays a critical role in the numerical study of nonlocal conservation laws.

## Abstract

We deal with the numerical investigation of the local limit of nonlocal conservation laws. Previous numerical experiments suggest convergence in the local limit. However, recent analytic results state that (i) in general convergence does not hold because one can exhibit counterexamples; (ii) convergence can be recovered provided viscosity is added to both the local and the nonlocal equations. Motivated by these analytic results, we investigate the role of numerical viscosity in the numerical study of the local limit of nonlocal conservation laws. In particular, we show that the numerical viscosity of Lax-Friedrichs type schemes jeopardizes the reliability of the numerical scheme and erroneously detects convergence in cases where convergence is ruled out by analytic results. We also test Godunov type schemes, less affected by numerical viscosity, and show that in some cases they provide more reliable results.