A stable absorbing boundary layer for anisotropic waves

TL;DR

This paper introduces a stable phase space filter boundary method for anisotropic wave models, addressing instability issues of traditional PML methods, and demonstrates its effectiveness on Euler and Maxwell equations.

Contribution

The paper presents a new phase space filter boundary method that is stable for all wave equations, including those where PML is unstable, with proven stability for anisotropic models.

Findings

The phase space filter is stable for anisotropic wave models.

The method effectively handles Euler and Maxwell equations.

Stability is theoretically proven.

Abstract

For some anisotropic wave models, the PML (perfectly matched layer) method of open boundaries can become polynomially or exponentially unstable in time. In this work we present a new method of open boundaries, the phase space filter, which is stable for all wave equations. Outgoing waves can be characterized as waves located near the boundary of the computational domain with group velocities pointing outward. The phase space filtering algorithm consists of applying a filter to the solution that removes outgoing waves only. The method presented here is a simplified version of the original phase space filter, originally described in [22] for the Schrodinger equation. We apply this method to anisotropic wave models for which the PML is unstable, namely the Euler equations (linearized about a constant jet flow) and Maxwell's equations in an anisotropic medium. Stability of the phase space filter is proved.