Error estimates for Gaussian beam superpositions
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
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.
High-frequency wave propagation, error estimates, Gaussian beams
IdentifiersURN: urn:nbn:se:kth:diva-134760DOI: 10.1090/S0025-5718-2012-02656-1ISI: 000326287500012ScopusID: 2-s2.0-84873266486OAI: oai:DiVA.org:kth-134760DiVA: diva2:668181
FunderSwedish e‐Science Research Center
QC 201311292013-11-292013-11-282013-12-03Bibliographically approved