TY - JOUR

T1 - Operator-Splitting Based Fast Sweeping Methods for Isotropic Wave Propagation in a Moving Fluid

AU - Glowinski, Roland

AU - Leung, Shingyu

AU - Qian, Jianliang

N1 - Funding Information:
Department of Mathematics, University of Houston, Houston, TX 77204, and Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong (roland@math.uh.edu). This author’s research was supported by NSF grant DMS 1418308 and the Institute for Advanced Studies at the Hong Kong University of Science and Technology.
Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong (masyleung@ust.hk). This author’s research was supported by the Hong Kong RGC under grants 605612 and 16303114.
Department of Mathematics, Michigan State University, East Lansing, MI 48824 (qian@math. msu.edu). This author’s research was supported by NSF grants DMS 1222368 and 1522249.
Publisher copyright:
© 2016, Society for Industrial and Applied Mathematics

PY - 2016/4/26

Y1 - 2016/4/26

N2 - Wave propagation in an isotropic acoustic medium occupied by a moving fluid is governed by an anisotropic eikonal equation. Since this anisotropic eikonal equation is associated with an inhomogeneous Hamiltonian, most of existing anisotropic eikonal solvers are either inapplicable or of unpredictable behavior in convergence. Realizing that this anisotropic eikonal equation is defined by a sum of two well-understood first-order differential operators, we propose novel operator-splitting based fast sweeping methods to solve this generalized eikonal equation. We develop various operator-splitting methods relying on the Peaceman--Rachford scheme, the Douglas--Rachford scheme, the θ-scheme, and the regularized θ-scheme. After applying the operator-splitting strategy, each splitting step corresponds to a much simpler Hamilton--Jacobi equation so that we can apply the Lax--Friedrichs sweeping method to solve these splitted equations efficiently and easily. Two- and three-dimensional examples demonstrate the performances and efficiency of the proposed approaches.

AB - Wave propagation in an isotropic acoustic medium occupied by a moving fluid is governed by an anisotropic eikonal equation. Since this anisotropic eikonal equation is associated with an inhomogeneous Hamiltonian, most of existing anisotropic eikonal solvers are either inapplicable or of unpredictable behavior in convergence. Realizing that this anisotropic eikonal equation is defined by a sum of two well-understood first-order differential operators, we propose novel operator-splitting based fast sweeping methods to solve this generalized eikonal equation. We develop various operator-splitting methods relying on the Peaceman--Rachford scheme, the Douglas--Rachford scheme, the θ-scheme, and the regularized θ-scheme. After applying the operator-splitting strategy, each splitting step corresponds to a much simpler Hamilton--Jacobi equation so that we can apply the Lax--Friedrichs sweeping method to solve these splitted equations efficiently and easily. Two- and three-dimensional examples demonstrate the performances and efficiency of the proposed approaches.

KW - Anisotropic eikonal equation

KW - Fast sweeping methods

KW - Higher-order schemes

KW - Operator-splitting methods

UR - http://www.scopus.com/inward/record.url?scp=84964855642&partnerID=8YFLogxK

U2 - 10.1137/15M1043868

DO - 10.1137/15M1043868

M3 - Article

AN - SCOPUS:84964855642

SN - 1064-8275

VL - 38

SP - A1195-A1223

JO - SIAM Journal of Scientific Computing

JF - SIAM Journal of Scientific Computing

IS - 2

ER -