Efficient PML for the wave equation

TL;DR

This paper introduces a simple, efficient PML formulation directly for the second-order wave equation, reducing computational complexity and enabling easy integration with standard numerical methods, while maintaining stability and accuracy.

Contribution

A novel PML formulation for the wave equation in second-order form that requires fewer auxiliary variables and is straightforward to implement and couple with existing numerical methods.

Findings

Requires only two auxiliary variables in 2D and four in 3D.

Proves strong stability of the formulation.

Numerical examples demonstrate accuracy and long-term stability.

Abstract

In the last decade, the perfectly matched layer (PML) approach has proved a flexible and accurate method for the simulation of waves in unbounded media. Most PML formulations, however, usually require wave equations stated in their standard second-order form to be reformulated as first-order systems, thereby introducing many additional unknowns. To circumvent this cumbersome and somewhat expensive step, we instead propose a simple PML formulation directly for the wave equation in its second-order form. Inside the absorbing layer, our formulation requires only two auxiliary variables in two space dimensions and four auxiliary variables in three space dimensions; hence it is cheap to implement. Since our formulation requires no higher derivatives, it is also easily coupled with standard finite difference or finite element methods. Strong stability is proved while numerical examples in two and three space dimensions illustrate the accuracy and long time stability of our PML formulation.