Fast Multiscale Gaussian Beam Method for Three-Dimensional Elastic Wave Equations in Bounded Domains
Jianliang Qian, Chao Song · SIAM Journal on Numerical Analysis · 2021
We propose a new fast multiscale Gaussian beam method to solve the three-dimensional elastic wave equation in a bounded domain in the high-frequency regime. We develop a novel multiscale transform to decompose an arbitrary vector-valued function into multiple Gaussian wavepackets with various resolutions. We consider both periodic and Dirichlet boundary conditions, and we further derive various reflection rules to compute crucial multiscale Gaussian beam ingredients so as to enforce these boundary conditions. To improve efficiency and accuracy of multiscale beam propagation, we develop a new reinitialization strategy based on the stationary phase approximation so that we can sharpen each single beam. Such a reinitialization strategy is especially useful and necessary to treat the shear-wave reflection. Numerical examples in different setups demonstrate correctness and robustness of the new method. We also show numerically that the convergence rate of the proposed multiscale Gaussian beam method follows that of the classical Gaussian beam solution, i.e., $O\big(\frac{1}{\sqrt{\omega}}\big)$, where $\omega$ is the largest frequency in the underlying wave motion.