On the Quasi-Random Choice Method for the Liouville Equation of Geometrical Optics with Discontinuous Wave Speed

Abstract

We study the quasi-random choice method (QRCM) for the Liouville equation of geometrical optics with discontinuous local wave speed. This equation arises in the phase space computation of high frequency waves through interfaces, where waves undergo partial transmissions and reections. The numerical challenges include interface, contact discontinuities, and measure-valued solutions. The so-called QRCM is a random choice method based on quasi-random sampling (a deterministic alternative to random sampling). The method not only is viscosity-free but also provides faster convergence rate. Therefore, it is appealing for the problem under study which is indeed a Hamiltonian ow. Our analysis and computational results show that the QRCM 1) is almost rst-order accurate even with the aforementioned discontinuities; 2) gives sharp resolutions for all discontinuities encountered in the problem; and 3) for measure-valued solutions, does not need the level set decomposition for nite dierence/volume methods with numerical viscosities.