### Hamiltonian-Preserving Schemes for the Liouville Equation of Geometrical Optics with Transmissions and Reflections

Shi Jin and Xin Wen

Abstract: We construct a class of Hamiltonian-preserving numerical schemes for a Liouville equation of geometrical optics, with transmissions and reflections. This equation arises in the high frequency limit of the linear wave equation, with discontinuous local wave speeds. In our previous work 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 coexistence of both transmission and reflection satisfying the Snell 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.

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Author(s):
Shi Jin, <jin@math.wisc.edu>
Xin Wen, <wenx@mail.tsinghua.edu.cn>
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