Volume 31, Issue 6
On the Quasi-Random Choice Method for the Liouville Equation of Geometrical Optics with Discontinuous Wave Speed

Jingwei Hu & Shi Jin

J. Comp. Math., 31 (2013), pp. 573-591.

Published online: 2013-12

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  • 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 re ections. 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 di erence/volume methods with numerical viscosities.

  • Keywords

Liouville equation High frequency wave Interface Measure-valued solution Random choice method Quasi-random sequence

  • AMS Subject Headings

35L45 65M06 11K45.

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COPYRIGHT: © Global Science Press

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@Article{JCM-31-573, author = {Jingwei Hu and Shi Jin}, title = {On the Quasi-Random Choice Method for the Liouville Equation of Geometrical Optics with Discontinuous Wave Speed}, journal = {Journal of Computational Mathematics}, year = {2013}, volume = {31}, number = {6}, pages = {573--591}, 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 re ections. 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 di erence/volume methods with numerical viscosities.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1309-m3592}, url = {http://global-sci.org/intro/article_detail/jcm/9755.html} }
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 KW - High frequency wave KW - Interface KW - Measure-valued solution KW - Random choice method KW - Quasi-random sequence AB -

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 re ections. 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 di erence/volume methods with numerical viscosities.

Jingwei Hu & Shi Jin. (1970). On the Quasi-Random Choice Method for the Liouville Equation of Geometrical Optics with Discontinuous Wave Speed. Journal of Computational Mathematics. 31 (6). 573-591. doi:10.4208/jcm.1309-m3592
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