Numerical simulation of singularity propagation modeled by linear convection equations with spatially heterogeneous nonlocal interactions

TL;DR

This paper investigates how singularities propagate in solutions of linear convection equations with spatially varying nonlocal interactions, combining analytical derivations and numerical simulations to understand the effects of heterogeneity.

Contribution

It introduces a model with a spatially varying nonlocal horizon and develops analytical and numerical methods to study singularity propagation in heterogeneous nonlocal systems.

Findings

Singularity propagation is significantly affected by nonlocal horizon heterogeneity.

Analytical equations characterize different singularity types in nonlocal regimes.

Numerical simulations illustrate diverse propagation patterns under heterogeneity.

Abstract

We study the propagation of singularities in solutions of linear convection equations with spatially heterogeneous nonlocal interactions. A spatially varying nonlocal horizon parameter is adopted in the model, which measures the range of nonlocal interactions. Via heterogeneous localization, this can lead to the seamless coupling of the local and nonlocal models. We are interested in understanding the impact on singularity propagation due to the heterogeneities of nonlocal horizon and the local and nonlocal transition. We first analytically derive equations to characterize the propagation of different types of singularities for various forms of nonlocal horizon parameters in the nonlocal regime. We then use asymptotically compatible schemes to discretize the equations and carry out numerical simulations to illustrate the propagation patterns in different scenarios.