A discontinuous Galerkin method for one-dimensional nonlocal wave problems

TL;DR

This paper introduces a fully discrete discontinuous Galerkin scheme for one-dimensional nonlocal wave equations, providing rigorous error analysis, energy preservation, and asymptotic compatibility with classical wave equations.

Contribution

It develops a novel DG-based numerical scheme with theoretical error bounds, energy conservation, and asymptotic compatibility for nonlocal wave problems.

Findings

Optimal L2 error convergence proven for specific kernels

Scheme preserves energy of the nonlocal wave equation

Asymptotic compatibility with local wave equations demonstrated

Abstract

This paper presents a fully discrete numerical scheme for one-dimensional nonlocal wave equations and provides a rigorous theoretical analysis. To facilitate the spatial discretization, we introduce an auxiliary variable analogous to the gradient field in local discontinuous Galerkin (DG) methods for classical partial differential equations (PDEs) and reformulate the equation into a system of equations. The proposed scheme then uses a DG method for spatial discretization and the Crank-Nicolson method for time integration. We prove optimal L2 error convergence for both the solution and the auxiliary variable under a special class of radial kernels at the semi-discrete level. In addition, for general kernels, we demonstrate the asymptotic compatibility of the scheme, ensuring that it recovers the classical DG approximation of the local wave equation in the zero-horizon limit. Furthermore, we prove that the fully discrete scheme preserves the energy of the nonlocal wave equation. A series of numerical experiments are presented to validate the theoretical findings.