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Sobolev and max norm error estimates for Gaussian beam superpositions
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). KTH, Centres, SeRC - Swedish e-Science Research Centre.ORCID iD: 0000-0002-6321-8619
2016 (English)In: Communications in Mathematical Sciences, ISSN 1539-6746, E-ISSN 1945-0796, Vol. 14, no 7, p. 2037-2072Article in journal (Refereed) Published
Abstract [en]

This work is concerned with the accuracy of Gaussian beam superpositions, which are asymptotically valid high frequency solutions to linear hyperbolic partial differential equations and the Schrödinger equation. We derive Sobolev and max norms estimates for the difference between an exact solution and the corresponding Gaussian beam approximation, in terms of the short wavelength e. The estimates are performed for the scalar wave equation and the Schrödinger equation. Our result demonstrates that a Gaussian beam superposition with kth order beams converges to the exact solution as O(εk/2-s) in order s Sobolev norms. This result is valid in any number of spatial dimensions and it is unaffected by the presence of caustics in the solution. In max norm, we show that away from caustics the convergence rate is O(ε⌈k/2⌉) and away from the essential support of the solution, the convergence is spectral in ε. However, in the neighborhood of a caustic point we are only able to show the slower, and dimensional dependent, rate O(ε(k-n)/2) in n spatial dimensions.

Place, publisher, year, edition, pages
International Press of Boston , 2016. Vol. 14, no 7, p. 2037-2072
Keywords [en]
Error estimates, Gaussian beams, High-frequency wave propagation, Max norm, Sobolev norm
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-195136DOI: 10.4310/CMS.2016.v14.n7.a12ISI: 000385990500012Scopus ID: 2-s2.0-84990954990OAI: oai:DiVA.org:kth-195136DiVA, id: diva2:1044832
Funder
Swedish e‐Science Research Center
Note

QC 20161107

Available from: 2016-11-07 Created: 2016-11-02 Last updated: 2017-11-29Bibliographically approved

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Runborg, Olof

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