Computation of transmissions and reflections in geometrical optics via the reduced Liouville equation

We develop a numerical scheme for the wave front computation of complete transmissions and reflections in geometrical optics. Such a problem can be formulated by a reduced Liouville equation with a discontinuous local wave speed or index of refraction, arising in the high frequency limit of linear waves through inhomogeneous media. The key idea is to incorporate Snell's Law of Refraction into the numerical flux for the reduced Liouville equation. This scheme allows a hyperbolic CFL condition, under which positivity, and stabilities in both l∞ and l1 norms, are established. Numerical experiments are carried out to demonstrate the validity and accuracy of this new scheme.