
Minimax Theory and its Applications 08 (2023), No. 1, 171212 Copyright Heldermann Verlag 2023 Truncated HadamardBabich Ansatz and Fast Huygens Sweeping Methods for TimeHarmonic Elastic Wave Equations in Inhomogeneous Media in the Asymptotic Regime Jianliang Qian Department of Mathematics, Michigan State University, East Lansing, U.S.A. qian@math.msu.edu Jian Song Department of Mathematics, Michigan State University, East Lansing, U.S.A. songji12@msu.edu Wangtao Lu School of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang, China wangtaolu@zju.edu.cn Robert Burridge Dept. of Mathematics and Statistics, University of New Mexico, Albuquerque, U.S.A. burridge137@gmail.com In some applications, it is reasonable to assume that geodesics (rays) have a consistent orientation so that a timeharmonic 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 [HadamardBabich ansatz for pointsource elastic wave equations in variable media at high frequencies, Multiscale Model Simul. 19/1 (2021) 4686], we propose a new truncated HadamardBabich ansatz based globally valid asymptotic method, dubbed the fast Huygens sweeping method, for computing Green's functions of frequencydomain pointsource elastic wave equations in inhomogeneous media in the highfrequency asymptotic regime and in the presence of caustics. The first novelty of the fast Huygens sweeping method is that the HuygensKirchhoff secondarysource 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 matrixvector products induced by the HuygensKirchhoff 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. Threedimensional numerical examples are presented to demonstrate the performance and accuracy of the new method. Keywords: HadamardBabich ansatz, elastic wave, eikonal equation, fast Huygens sweeping, butterfly algorithm. MSC: 78A05, 78A46, 78M35. [ Fulltextpdf (2721 KB)] for subscribers only. 