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

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 the phase space description of 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 in [S. Jin, X. Wen, Hamiltonian-preserving schemes for the Liouville equation with discontinuous potentials, Commun. Math. Sci. 3 (2005) 285–315] 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 a flat interface, the Hamiltonian preservation is shown to be 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 l1 and l∞ norms, are established for both schemes. The approach also provides a selection criterion for a unique solution of the underlying linear hyperbolic equation with singular (discontinuous and measure-valued) coefficients. Benchmark numerical examples are given, with analytic solution constructed, to study the numerical accuracy of these schemes.