### Hamiltonian-Preserving Schemes for the Liouville Equation of Geometrical Optics with Discontinuous Local Wave Speeds

Shi Jin and Xin Wen

Abstract: In this paper we construct two classes of Hamiltonian-preserving numerical schemes for a Liouville equation with discontinuous local wave speed. This equation arises in phase space description of the geometrical optics, and has been the foundation of the recently developed level set methods for multivalued solution in geometrical optics. We extend our previous work for the semiclassical limit of the Schrödinger equation into this system. The designing principle of the Hamiltonian preservation by building in the particle behavior at the interface into the numerical flux is used here, and as a consequence we obtain two classes of schemes that allow a hyperbolic stability condition. When a plane wave hits an interface, the Hamiltonian preservation is equivalent to Snell's law of refraction in the case when the ratio of wave length over the width of the interface goes to zero, when both length scales go to zero. Positivity, and stabilities in both $l^1$ and $l^\infty$ norms, are established for both schemes. The approach also provides a selection criterion for a unique weak solution of the underlying linear hyperbolic equations with singular coefficients. 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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