We consider the scalar anisotropic wave equation. Recently a convergence analysis for radial perfectly matched layers (PML) in the frequency domain was reported and in the present article we continue this approach into the time domain. First we explain why there is a good hope that radial complex scalings can overcome the instabilities of PML methods caused by anisotropic materials. Next we discuss some sensitive details, which seem like a paradox at the first glance: if the absorbing layer and the inhomogeneities are sufficiently separated, then the solution is indeed stable. However, for more general data the problem becomes unstable. In numerical computations we observe instabilities regardless of the position of the inhomogeneities, although the instabilities arise only for fine enough discretizations. As a remedy we propose a complex frequency shifted scaling and discretizations by Hardy space infinite elements or truncation-free PMLs. We show numerical experiments which confirm the stability and convergence of these methods.

翻译：本文考虑标量各向异性波动方程。近期已有研究报道了频域中径向完美匹配层（PML）的收敛性分析，本文则将该方法推广至时域。首先，我们阐释为何径向复伸缩有望克服各向异性材料引发的PML方法不稳定性。随后讨论若干敏感细节——初看时似乎存在矛盾：若吸收层与不均匀体充分分离，解确实稳定；但对于更一般的数据，问题将变得不稳定。数值计算中，无论不均匀体位置如何，均观测到不稳定性，尽管这种不稳定性仅出现在足够精细的离散化情形下。作为解决方案，我们提出复频移伸缩及基于Hardy空间无限元或无截断PML的离散化方法。数值实验验证了这些方法的稳定性与收敛性。