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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Error estimates for Gaussian beam superpositions


Authors: Hailiang Liu, Olof Runborg and Nicolay M. Tanushev
Journal: Math. Comp. 82 (2013), 919-952
MSC (2010): Primary 35J10, 35L05, 35A35, 41A60, 35L30
Published electronically: November 6, 2012
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Abstract: 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 Schrödinger 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 $ \varepsilon $. We present a systematic construction of Gaussian beam superpositions for all strictly hyperbolic and Schrödinger equations subject to highly oscillatory initial data of the form $ Ae^{i\Phi /\varepsilon }$. 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 $ \varepsilon ^{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, $ \varepsilon $-scaled, energy norm and for the Schrödinger 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 $ \mathbb{R}^2$ to analyze the sharpness of the theoretical results.


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Additional Information

Hailiang Liu
Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011
Email: hliu@iastate.edu

Olof Runborg
Affiliation: Department of Numerical Analysis, CSC, KTH, 100 44 Stockholm, Sweden – and – Swedish e-Science Research Center (SeRC), KTH, 100 44 Stockholm, Sweden
Email: olofr@nada.kth.se

Nicolay M. Tanushev
Affiliation: Department of Mathematics, The University of Texas at Austin, 1 University Station, C1200, Austin, Texas 78712
Address at time of publication: Z-Terra Inc., 17171 Park Row, Suite 247, Houston, Texas 77084
Email: nicktan@math.utexas.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-2012-02656-1
PII: S 0025-5718(2012)02656-1
Keywords: High-frequency wave propagation, error estimates, Gaussian beams
Received by editor(s): August 6, 2010
Received by editor(s) in revised form: June 2, 2011, and August 31, 2011
Published electronically: November 6, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.