Gaussian Beam ansatz for finite difference wave equations

TL;DR

This paper develops a rigorous Gaussian Beam ansatz for finite difference wave equations, addressing the challenges of high-frequency numerical solutions and their propagation properties, supported by numerical validation.

Contribution

It introduces a novel, rigorous Gaussian Beam construction tailored for semi-discretized wave equations, bridging the gap between continuous and numerical wave propagation behaviors.

Findings

Successful construction of Gaussian Beam solutions for finite difference schemes

Numerical simulations confirm accurate wave propagation modeling

Identification of high-frequency behavior differences between continuous and discrete settings

Abstract

This work is concerned with the construction of Gaussian Beam (GB) solutions for the numerical approximation of wave equations, semi-discretized in space by finite difference schemes. GB are high-frequency solutions whose propagation can be described, both at the continuous and at the semi-discrete levels, by microlocal tools along the bi-characteristics of the corresponding Hamiltonian. Their dynamics differ in the continuous and the semi-discrete setting, because of the high-frequency gap between the Hamiltonians. In particular, numerical high-frequency solutions can exhibit spurious pathological behaviors, such as lack of propagation in space, contrary to the classical space-time propagation properties of continuous waves. This gap between the behavior of continuous and numerical waves introduces also significant analytical difficulties, since classical GB constructions cannot be immediately extrapolated to the finite difference setting, and need to be properly tailored to accurately detect the propagation properties in discrete media. Our main objective in this paper is to present a general and rigorous construction of the GB ansatz for finite difference wave equations, and corroborate this construction through accurate numerical simulations.