Convergence of the Frozen Gaussian Approximation for High Frequency Elastic Waves

TL;DR

This paper extends the frozen Gaussian approximation (FGA) to elastic wave equations, demonstrating its convergence despite the challenges posed by repeated eigenvalues and wave coupling in high frequency elastic wave modeling.

Contribution

It derives the FGA for elastic waves, introduces a strong form for the evolution equations, and proves convergence using new energy estimates.

Findings

Convergence of FGA for elastic wave equations established.

Identification of diabatic coupling between SH and SV wave amplitudes.

Extension of FGA applicability to more complex hyperbolic systems.

Abstract

The frozen Gaussian approximation (FGA) is an effective tool for modeling high frequency wave propagation. In previous works, the convergence of the FGA has established for strict hyperbolic systems. In this work, we derive the frozen Gaussian approximation for the elastic wave equation, which can be cast as a hyperbolic system with repeated eigenvalues. In the derivation, the strong form for the evolution equation is introduced. A diabatic coupling is observed for the amplitude of the evolution equations between the SH, SV waves. Using previous results with new energy estimates we establish the convergence for the first order FGA for the elastic wave equation.