Computational high frequency waves through curved interfaces via the Liouville equation and geometric theory of diffraction

Authors:
Shi Jin;Dongsheng Yin
Affiliations:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, PR China and Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA;Department of Mathematical Sciences and the Center for Advanced Study, Tsinghua University, Beijing 100084, PR China
Venue:
Journal of Computational Physics
Year:
2008

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Abstract

We construct a class of numerical schemes for the Liouville equation of geometric optics coupled with the Geometric Theory of Diffractions to simulate the high frequency linear waves with a discontinuous index of refraction. In this work [S. Jin, X. Wen, A Hamiltonian-preserving scheme for the Liouville equation of geometric optics with partial transmissions and reflections, SIAM J. Numer. Anal. 44 (2006) 1801-1828], a Hamiltonian-preserving scheme for the Liouville equation was constructed to capture partial transmissions and reflections at the interfaces. This scheme is extended by incorporating diffraction terms derived from Geometric Theory of Diffraction into the numerical flux in order to capture diffraction at the interface. We give such a scheme for curved interfaces. This scheme is proved to be positive under a suitable time step constraint. Numerical experiments show that it can capture diffraction phenomena without fully resolving the wave length of the original wave equation.