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Error estimates for Gaussian beam superpositions
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.ORCID iD: 0000-0002-6321-8619
2013 (English)In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 82, no 282, 919-952 p.Article in journal (Refereed) Published
Abstract [en]

Gaussian beams are asymptotically valid high frequency solutions to hyperbolic partial differential equations, concentrated on a single curve through the physical domain. They can also be extended to some dispersive wave equations, such as the Schrodinger equation. Superpositions of Gaussian beams provide a powerful tool to generate more general high frequency solutions that are not necessarily concentrated on a single curve. This work is concerned with the accuracy of Gaussian beam superpositions in terms of the wavelength epsilon. We present a systematic construction of Gaussian beam superpositions for all strictly hyperbolic and Schrodinger equations subject to highly oscillatory initial data of the form Ae(i Phi/) (epsilon). Through a careful estimate of an oscillatory integral operator, we prove that the k-th order Gaussian beam superposition converges to the original wave field at a rate proportional to epsilon(k/2) in the appropriate norm dictated by the well-posedness estimate. In particular, we prove that the Gaussian beam superposition converges at this rate for the acoustic wave equation in the standard, epsilon-scaled, energy norm and for the Schrodinger equation in the L-2 norm. The obtained results are valid for any number of spatial dimensions and are unaffected by the presence of caustics. We present a numerical study of convergence for the constant coefficient acoustic wave equation in R-2 to analyze the sharpness of the theoretical results.

Place, publisher, year, edition, pages
2013. Vol. 82, no 282, 919-952 p.
Keyword [en]
High-frequency wave propagation, error estimates, Gaussian beams
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-134760DOI: 10.1090/S0025-5718-2012-02656-1ISI: 000326287500012Scopus ID: 2-s2.0-84873266486OAI: oai:DiVA.org:kth-134760DiVA: diva2:668181
Funder
Swedish e‐Science Research Center
Note

QC 20131129

Available from: 2013-11-29 Created: 2013-11-28 Last updated: 2017-12-06Bibliographically approved

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Runborg, Olof

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