A Time-Dependent Phase Space Filter for Anisotropic Wave Equations on Unbounded Domains

Abstract. For wave propagation problems in homogeneous, isotropic media, exponentially convergent domain truncation techniques based on perfectly matched damping layers or optimal rational approximations to the exact radiation operator are known. However, these methods fail for systems with wave families satisfying general dispersion relations, forcing practitioners to resort to ad hoc procedures based on grid stretching and artificial damping. Here we propose a new method for constructing convergent approximations on truncated domains, the phase space filter, which unlike other methods is completely general and mathematically justified. Based on the fact that outgoing waves can be characterized as waves located near the boundary of the computational domain with group velocities pointing outward, the key idea of the phase space filtering algorithm consists of applying a filter to the solution that removes outgoing waves only. The method introduced in this work is a simplified version of the original phase space filter, which was proposed for the Schrödinger equation. The method is applied to anisotropic wave models for which existing techniques are unstable, namely free-space problems governed in the far field by the Euler equations linearized about a uniform mean flow and Maxwell’s equations in an anisotropic medium. We also present experiments in waveguide geometry and for isotropic problems which indicate that a more sophisticated multiscale extension of the algorithm is needed to obtain good long-time accuracy for waveguide problems, or to match the performance of the optimized local radiation conditions available in the isotropic case. Theoretical results concerning the convergence and computational costs of the phase space filter are discussed and stability is proven. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://zenodo.org/records/17237326 . [Formula: see text]