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Error estimates for Gaussian beam superpositions
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Numerisk analys, NA. KTH, Centra, SeRC - Swedish e-Science Research Centre.ORCID-id: 0000-0002-6321-8619
2013 (engelsk)Inngår i: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 82, nr 282, s. 919-952Artikkel i tidsskrift (Fagfellevurdert) 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.

sted, utgiver, år, opplag, sider
2013. Vol. 82, nr 282, s. 919-952
Emneord [en]
High-frequency wave propagation, error estimates, Gaussian beams
HSV kategori
Identifikatorer
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, id: diva2:668181
Forskningsfinansiär
Swedish e‐Science Research Center
Merknad

QC 20131129

Tilgjengelig fra: 2013-11-29 Laget: 2013-11-28 Sist oppdatert: 2017-12-06bibliografisk kontrollert

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