On the Quasi-Random Choice Method for the Liouville Equation of Geometrical Optics with Discontinuous Wave Speed (Journal of Computational Mathematics, 2021)
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 phasespace computation of high frequency waves through interfaces, where waves undergo partialtransmissions and reflections. The numerical challenges include interface, contact discontinuities, and measure-valued solutions. The so-called QRCM is a random choice methodbased on quasi-random sampling (a deterministic alternative to random sampling). Themethod not only is viscosity-free but also provides faster convergence rate. Therefore, itis appealing for the problem under study which is indeed a Hamiltonian flow. Our analysis and computational results show that the QRCM 1) is almost first-order accurate evenwith the aforementioned discontinuities; 2) gives sharp resolutions for all discontinuitiesencountered in the problem; and 3) for measure-valued solutions, does not need the levelset decomposition for finite difference/volume methods with numerical viscosities.