On the role of numerical viscosity in the study of the local limit of nonlocal conservation laws
For researchers studying nonlocal conservation laws, this work highlights the critical role of numerical viscosity in misleading numerical experiments, urging caution in interpreting numerical convergence.
The paper investigates how numerical viscosity in Lax-Friedrichs type schemes can falsely indicate convergence in the local limit of nonlocal conservation laws, where analytic results show convergence does not hold. Godunov type schemes, with less numerical viscosity, provide more reliable results.
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.