Recovery of high frequency wave fields for the acoustic wave equation

TL;DR

This paper develops a phase space based Gaussian beam superposition method for high frequency acoustic wave equations, proving convergence and demonstrating its effectiveness in capturing wave fields.

Contribution

It introduces a systematic phase space approach for constructing Gaussian beam superpositions and proves their convergence rate for high frequency solutions.

Findings

Superposition over two moving domains is necessary.

Convergence rate of Gaussian beam superpositions is established.

Method effectively captures high frequency wave fields.

Abstract

Computation of high frequency solutions to wave equations is important in many applications, and notoriously difficult in resolving wave oscillations. Gaussian beams are asymptotically valid high frequency solutions concentrated on a single curve through the physical domain, and superposition of Gaussian beams provides a powerful tool to generate more general high frequency solutions to PDEs. An alternative way to compute Gaussian beam components such as phase, amplitude and Hessian of the phase, is to capture them in phase space by solving Liouville type equations on uniform grids. Following \cite{LR:2009} we present a systematic construction of asymptotic high frequency wave fields from computations in phase space for acoustic wave equations; the superposition of phase space based Gaussian beams over two moving domains is shown necessary. Moreover, we prove that the $k$-th order Gaussian beam superposition converges to the original wave field in the energy norm, at the rate of $\ep^{\frac{k}{2}+\frac{1-n}{4}}$ in dimension $n$.