TY - JOUR
T1 - On the Quasi-Random Choice Method for the Liouville Equation of Geometrical Optics with Discontinuous Wave Speed
AU - Jingwei Hu & Shi Jin
JO - Journal of Computational Mathematics
VL - 6
SP - 573
EP - 591
PY - 2013
DA - 2013/12
SN - 31
DO - http://doi.org/10.4208/jcm.1309-m3592
UR - https://global-sci.org/intro/article_detail/jcm/9755.html
KW - Liouville equation, High frequency wave, Interface, Measure-valued solution,
Random choice method, Quasi-random sequence.
AB - <p style="text-align: justify;">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 reflections. 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 flow. Our analysis and computational results show that the QRCM 1) is almost first-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 finite difference/volume methods with numerical viscosities.</p>