On perfectly matched layers of nonlocal wave equations in unbounded multi-scale media

TL;DR

This paper develops a nonlocal perfectly matched layer (PML) for wave equations in unbounded multi-scale media, enabling effective wave absorption without derivatives and demonstrating high accuracy through numerical examples.

Contribution

It introduces a novel nonlocal PML construction and an asymptotically compatible scheme combining Talbot's contour and Verlet methods for efficient numerical solutions.

Findings

Wave decay exponentially in the PML layer

Reflections at boundary are minimized

Numerical examples confirm high accuracy

Abstract

A nonlocal perfectly matched layer (PML) is formulated for the nonlocal wave equation in the whole real axis and numerical discretization is designed for solving the reduced PML problem on a bounded domain. The nonlocal PML poses challenges not faced in PDEs. For example, there is no derivative in nonlocal models, which makes it impossible to replace derivates with complex ones. Here we provide a way of constructing the PML for nonlocal models, which decays the waves exponentially impinging in the layer and makes reflections at the truncated boundary very tiny. To numerically solve the nonlocal PML problem, we design the asymptotically compatible (AC) scheme for spatially nonlocal operator by combining Talbot's contour, and a Verlet-type scheme for time evolution. The accuracy and effectiveness of our approach are illustrated by various numerical examples.