Gaussian Beam ansatz for finite difference wave equations
Biccari U, Zuazua Iriondo E (2024)
Publication Language: English
Publication Status: Submitted
Publication Type: Unpublished / Preprint
Future Publication Type: Journal article
Publication year: 2024
URI: https://dcn.nat.fau.eu/wp-content/uploads/WaveAnsatz.pdf
Open Access Link: https://dcn.nat.fau.eu/wp-content/uploads/WaveAnsatz.pdf
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.
Authors with CRIS profile
Enrique Zuazua Iriondo
Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur)
Involved external institutions
Spain (ES)How to cite
APA:
Biccari, U., & Zuazua Iriondo, E. (2024). Gaussian Beam ansatz for finite difference wave equations. (Unpublished, Submitted).
MLA:
Biccari, Umberto, and Enrique Zuazua Iriondo. Gaussian Beam ansatz for finite difference wave equations. Unpublished, Submitted. 2024.
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