A Reflectionless Discrete Perfectly Matched Layer

TL;DR

This paper introduces a new discrete PML for wave simulations that completely eliminates numerical reflections, using Discrete Complex Analysis to derive an easily-implementable finite difference form.

Contribution

A novel discrete PML that produces zero numerical reflection, derived through Discrete Complex Analysis, with proven stability and practical implementation advantages.

Findings

No numerical reflection at the PML boundary.

Exponential damping of waves within the PML.

Error due to domain truncation is negligible.

Abstract

Perfectly Matched Layer (PML) is a widely adopted non-reflecting boundary treatment for wave simulations. Reducing numerical reflections from a discretized PML has been a long lasting challenge. This paper presents a new discrete PML for the multi-dimensional scalar wave equation which produces no numerical reflection at all. The reflectionless discrete PML is discovered through a straightforward derivation using Discrete Complex Analysis. The resulting PML takes an easily-implementable finite difference form with compact stencil. In practice, the discrete waves are damped exponentially in the PML, and the error due to domain truncation is maintained at machine zero by a moderately thick PML. The numerical stability of the proposed PML is also demonstrated.