Numerical solution of a one-dimensional nonlocal Helmholtz equation by Perfectly Matched Layers

TL;DR

This paper develops a nonlocal PML method for solving the one-dimensional nonlocal Helmholtz equation, providing stability analysis, exponential decay results, and numerical validation for the proposed approach.

Contribution

It extends PML techniques to nonlocal operators, derives stability estimates, and introduces an asymptotic compatibility scheme for truncated problems.

Findings

Weighted average of the nonlocal PML solution decays exponentially in PML layers.

Nonlocal Helmholtz solution decays exponentially outside a domain.

Numerical examples confirm the effectiveness of the nonlocal PML method.

Abstract

We consider the computation of a nonlocal Helmholtz equation by using Perfectly Matched Layer (PML). We first derive the nonlocal PML equation by extending PML modifications from the local operator to the nonlocal operator of integral form. We then give stability estimates of some weighted average value of the nonlocal Helmholtz solution and prove that (i) the weighted average value of the nonlocal PML solution decays exponentially in PML layers in one case; (ii) in the other case, the weighted average value of the nonlocal Helmholtz solution itself decays exponentially outside some domain. Particularly for a typical kernel function $\gamma_1(s)=\frac12 e^{-| s|}$, we obtain the Green's function of the nonlocal Helmholtz equation, and use the Green's function to further prove that (i) the nonlocal PML solution decays exponentially in PML layers in one case; (ii) in the other case, the nonlocal Helmholtz solution itself decays exponentially outside some domain. Based on our theoretical analysis, the truncated nonlocal problems are discussed and an asymptotic compatibility scheme is also introduced to solve the resulting truncated problems. Finally, numerical examples are provided to verify the effectiveness and validation of our nonlocal PML strategy and theoretical findings.