On the random choice method for the Liouville equation of geometrical optics with discontinuous local wave speeds∗

We study the random choice method (RCM) for the Liouville equation of geometrical optics with discontinuous local wave speeds. This problem arises in the computation of high frequency waves through interfaces, where waves undergo partial transmissions and reflections. The numerical challenges include interface, contact discontinuities, and measurevalued solutions. The purpose of this paper is to understand the performance of the RCM for this problem. Since the RCM does not contain numerical viscosity, it is appealing to use it for this problem which is indeed a Hamiltonian flow. Our analysis and computational results show that the RCM 1) is first order accurate even with the aforementioned discontinuities; 2) gives sharp resolutions for all discontinuities encountered in this problem; and 3) for measure-valued initial data and solutions, does not need to use the decomposition method proposed in [15] for finite difference or finite volume methods that contain numerical viscosities.