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Minimax Theory and its Applications 08 (2023), No. 1, 171--212
Copyright Heldermann Verlag 2023

Truncated Hadamard-Babich Ansatz and Fast Huygens Sweeping Methods for Time-Harmonic Elastic Wave Equations in Inhomogeneous Media in the Asymptotic Regime

Jianliang Qian
Department of Mathematics, Michigan State University, East Lansing, U.S.A.

Jian Song
Department of Mathematics, Michigan State University, East Lansing, U.S.A.

Wangtao Lu
School of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang, China

Robert Burridge
Dept. of Mathematics and Statistics, University of New Mexico, Albuquerque, U.S.A.

In some applications, it is reasonable to assume that geodesics (rays) have a consistent orientation so that a time-harmonic elastic wave equation may be viewed as an evolution equation in one of the spatial directions. With such applications in mind, motivated by our recent work [Hadamard-Babich ansatz for point-source elastic wave equations in variable media at high frequencies, Multiscale Model Simul. 19/1 (2021) 46--86], we propose a new truncated Hadamard-Babich ansatz based globally valid asymptotic method, dubbed the fast Huygens sweeping method, for computing Green's functions of frequency-domain point-source elastic wave equations in inhomogeneous media in the high-frequency asymptotic regime and in the presence of caustics. The first novelty of the fast Huygens sweeping method is that the Huygens-Kirchhoff secondary-source principle is used to integrate many locally valid asymptotic solutions to yield a globally valid asymptotic solution so that caustics can be treated automatically. This yields uniformly accurate solutions both near the source and away from it. The second novelty is that a butterfly algorithm is adapted to accelerate matrix-vector products induced by the Huygens-Kirchhoff integral.
The new method enjoys the following desired features: (1) it treats caustics automatically; (2) precomputed asymptotic ingredients can be used to construct Green's functions of elastic wave equations for many different point sources and for arbitrary frequencies; (3) given a specified number of points per wavelength, it constructs Green's functions in nearly optimal complexity O(N log N) in terms of the total number of mesh points N, where the prefactor of the complexity depends only on the specified accuracy and is independent of the frequency parameter. Three-dimensional numerical examples are presented to demonstrate the performance and accuracy of the new method.

Keywords: Hadamard-Babich ansatz, elastic wave, eikonal equation, fast Huygens sweeping, butterfly algorithm.

MSC: 78A05, 78A46, 78M35.

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