General superpositions of Gaussian beams and propagation errors

2019/05/22 by Liu, Hailiang, Ralston, James, Yin, Peimeng
#35A35 #35L05 #41A60 #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1905.09090

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 [ H. Liu, O. Runborg, and N. M. Tanushev, Math. Comp., 82: 919--952, 2013] can be applied to obtain the propagation error of order k^{1- (N)/(2)- (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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