On the convergence of frozen Gaussian approximation for linear non-strictly hyperbolic systems

Emmanuel Lorin (School of Mathematics and Statistics, Carleton University, Ottawa, Canada; and Centre de Recherches Mathématiques, Université de Montréal, Canada)

Xu Yang (Department of Mathematics, University of California, Santa Barbara, Calif., U.S.A.)

Abstract

Frozen Gaussian approximation (FGA) has been applied and numerically verified as an efficient tool to compute high-frequency wave propagation modeled by non-strictly hyperbolic systems, such as the elastic wave equations [J.C. Hateley, L. Chai, P. Tong and X. Yang, Geophys. J. Int., 216:1394–1412, 2019] and the Dirac system [L. Chai, E. Lorin and X. Yang, SIAM J. Numer. Anal., 57:2383–2412, 2019]. However, the theory of convergence is still incomplete for non-strictly hyperbolic systems, where the latter can be interpreted as a diabatic (or more) coupling. In this paper, we establish the convergence theory for FGA for linear non-strictly hyperbolic systems, with an emphasis on the elastic wave equations and the Dirac system. Unlike the convergence theory of FGA for strictly linear hyperbolic systems, the key estimate lies in the boundedness of intraband transitions in diabatic coupling.

Keywords

frozen Gaussian approximation, convergence, non-strictly hyperbolic, elastic wave equations, Dirac equation

2010 Mathematics Subject Classification

65M12, 81Q05

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L.C. was partially supported by the NSFC grant 11901601. J.C.H. and X.Y. were partially supported by the NSF grant DMS-1818592. E.L. was partially supported through the NSERC Discovery Grant program.

Received 17 April 2020

Accepted 21 September 2020

Published 5 May 2021