Asymptotically compatible approximations of linear nonlocal conservation laws with variable horizon

Abstract

AbstractWe consider a quadrature‐based finite difference discretization of one‐dimensional scalar linear nonlocal conservation laws. The range of nonlocal interactions is allowed to vary in the spatial domain. We are particularly concerned with the convergence of the discrete approximation both in the nonlocal setting and in the local limit as the horizon parameter approaches zero. We present the first complete proof of the convergence of numerical discretization to both the nonlocal regime and the local limit of all feasible kernels, which in particular, establishes the asymptotically compatibility of the numerical scheme. We also present numerical results to demonstrate the effect of the variable horizon on the wave propagation described by the nonlocal model.