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Mathematics of Computation

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General superpositions of Gaussian beams and propagation errors


Authors: Hailiang Liu, James Ralston and Peimeng Yin
Journal: Math. Comp. 89 (2020), 675-697
MSC (2010): Primary 35L05, 35A35, 41A60
DOI: https://doi.org/10.1090/mcom/3462
Published electronically: September 9, 2019
MathSciNet review: 4044446
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Abstract: Gaussian beams are asymptotically valid high frequency solutions concentrated on a single curve through the physical domain, and superposition of Gaussian beams provides a powerful tool to generate more general high frequency solutions to PDEs. We present a superposition of Gaussian beams over an arbitrary bounded set of dimension $ m$ in phase space, and show that the tools recently developed in [Math. Comp. 82 (2013), pp. 919-952] can be applied to obtain the propagation error of order $ k^{1- \frac {N}{2}- \frac {d-m}{4}}$, where $ N$ is the order of beams and $ d$ is the spatial dimension. Moreover, we study the sharpness of this estimate in examples.


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Additional Information

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

James Ralston
Affiliation: Department of Mathematics, UCLA, Los Angeles, California 90095
Email: ralston@math.ucla.edu

Peimeng Yin
Affiliation: Departmetn of Mathematics, Iowa State University, Ames, Iowa 50011
Email: pemyin@iastate.edu

DOI: https://doi.org/10.1090/mcom/3462
Keywords: High frequency wave propagation, Gaussian beams, phase space, superposition, error estimates
Received by editor(s): August 29, 2019
Received by editor(s) in revised form: February 21, 2019, and April 9, 2019
Published electronically: September 9, 2019
Additional Notes: This work was supported by the National Science Foundation under Grant RNMS (Ki-Net) 1107291 and by NSF Grant DMS1812666.
Article copyright: © Copyright 2019 American Mathematical Society