Communications in Mathematical Sciences

Volume 13 (2015)

Number 4

Special Issue in Honor of George Papanicolaou’s 70th Birthday

Guest Editors: Liliana Borcea, Jean-Pierre Fouque, Shi Jin, Lenya Ryzhik, and Jack Xin

Computing high frequency solutions of symmetric hyperbolic systems with polarized waves

Pages: 1001 – 1024

DOI: https://dx.doi.org/10.4310/CMS.2015.v13.n4.a8

Authors

Leland Jefferis (Department of Mathematics, University of Wisconsin, Madison, Wisc., U.S.A.)

Shi Jin (Dept. of Mathematics, Institute of Natural Sciences and Ministry of Education, Key Lab of Scientific and Engineering Computing, Shanghai Jiao Tong University, Shanghai, China; and Dept. of Mathematics, University of Wisconsin, Madison, Wisc., U.S.A.)

Abstract

We develop computational methods for high frequency solutions of general symmetric hyperbolic systems with eigenvalue degeneracies (multiple eigenvalues with constant multiplicities) in the dispersion matrices that correspond to polarized waves. Physical examples of such systems include the three-dimensional elastic waves and Maxwell equations. The computational methods are based on solving a coupled system of inhomogeneous Liouville equations which is the high frequency limit of the underlying hyperbolic systems by using the Wigner transform [L. Ryzhik, G. Papanicolaou, and J. Keller, Wave Motion, 24(4), 327–370, 1996]. We first extend the level set methods developed in [S. Jin, H. Liu, S. Osher, and R. Tsai, Journal of Computational Physics, 210, 497–518, 2005] for the homogeneous Liouville equation to the coupled inhomogeneous system, and find an efficient simplification in one space dimension for the Eulerian formulation which reduces the computational cost of two-dimensional phase space Liouville equations into that of two one-dimensional equations. For the Lagrangian formulation, we introduce a geometric method which allows a significant simplification in the numerical evaluation of the energy density and flux. Numerical examples are presented in both one and two space dimensions to demonstrate the validity of the methods in the high frequency regime.

Keywords

Gaussian beams methods, high frequency waves

2010 Mathematics Subject Classification

00A69, 74J05

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Published 12 March 2015