On the non-equivalence of perfectly matched layers and exterior complex scaling

TL;DR

This paper compares PML and ECS absorbing boundary methods, revealing their fundamental mathematical differences and distinct impacts on spectral properties and stability through spectral analysis and numerical examples.

Contribution

It demonstrates that PML and ECS are mathematically and numerically distinct, with different spectral effects and stability characteristics, challenging the assumption of their equivalence.

Findings

ECS rotates the spectrum into the complex plane.

PML shifts the spectrum via a complex gauge transform.

The methods differ in time-stability properties.

Abstract

The perfectly matched layers (PML) and exterior complex scaling (ECS) methods for absorbing boundary conditions are analyzed using spectral decomposition. Both methods are derived through analytical continuations from unitary to contractive transformations. We find that the methods are mathematically and numerically distinct: ECS is complex stretching that rotates the operator's spectrum into the complex plane, whereas PML is a complex gauge transform which shifts the spectrum. Consequently, the schemes differ in their time-stability. Numerical examples are given.