A Hamiltonian-Preserving Scheme for the Liouville Equation of Geometrical Optics with Partial Transmissions and Reflections

Authors:
Shi Jin;Xin Wen
Affiliations:
-;-
Venue:
SIAM Journal on Numerical Analysis
Year:
2006

Citing 0
Cited 6

Hamiltonian-preserving schemes for the Liouville equation of geometrical optics with discontinuous local wave speeds

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High Order Numerical Methods to Three Dimensional Delta Function Integrals in Level Set Methods

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Abstract

We construct a class of Hamiltonian-preserving numerical schemes for the Liouville equation of geometrical optics, with partial transmissions and reflections. This equation arises in the high frequency limit of the linear wave equation, with a discontinuous index of refraction. In our previous work [Hamiltonian-preserving schemes for the Liouville equation of geometrical optics with discontinuous local wave speeds, J. Comput. Phys. 214 (2006), pp. 672-697], we introduced the Hamiltonian-preserving schemes for the same equation when only complete transmissions or reflections occur at the interfaces. These schemes are extended in this paper to the general case of partial transmissions and reflections. The key idea is to build into the numerical flux the behavior of waves at the interface, namely, partial transmissions and reflections that satisfy Snell’s law of refraction with the correct transmission and reflection coefficients. This scheme allows a hyperbolic stability condition, under which positivity, and stabilities in both $l^1$ and $l^\infty$ norms, are established. Numerical experiments are carried out to study the numerical accuracy.