Volume 1, Issue 5
Review Article: Multi-Valued Solution and Level Set Methods in Computational High Frequency Wave Propagation

H. Liu, S. Osher & R. Tsai

DOI:

Commun. Comput. Phys., 1 (2006), pp. 765-804.

Published online: 2006-01

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  • Abstract

We review the level set methods for computing multi-valued solutions to a class of nonlinear first order partial differential equations, including Hamilton-Jacobi equations, quasi-linear hyperbolic equations, and conservative transport equations with multi-valued transport speeds. The multivalued solutions are embedded as the zeros of a set of scalar functions that solve the initial value problems of a time dependent partial differential equation in an augmented space. We discuss the essential ideas behind the techniques, the coupling of these techniques to the projection of the interaction of zero level sets and a collection of applications including the computation of the semiclassical limit for Schrödinger equations and the high frequency geometrical optics limits of linear wave equations. 

  • Keywords

Multi-valued solution level set method high frequency wave propagation.

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@Article{CiCP-1-765, author = {}, title = {Review Article: Multi-Valued Solution and Level Set Methods in Computational High Frequency Wave Propagation}, journal = {Communications in Computational Physics}, year = {2006}, volume = {1}, number = {5}, pages = {765--804}, abstract = {

We review the level set methods for computing multi-valued solutions to a class of nonlinear first order partial differential equations, including Hamilton-Jacobi equations, quasi-linear hyperbolic equations, and conservative transport equations with multi-valued transport speeds. The multivalued solutions are embedded as the zeros of a set of scalar functions that solve the initial value problems of a time dependent partial differential equation in an augmented space. We discuss the essential ideas behind the techniques, the coupling of these techniques to the projection of the interaction of zero level sets and a collection of applications including the computation of the semiclassical limit for Schrödinger equations and the high frequency geometrical optics limits of linear wave equations. 

}, issn = {1991-7120}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/cicp/7979.html} }
TY - JOUR T1 - Review Article: Multi-Valued Solution and Level Set Methods in Computational High Frequency Wave Propagation JO - Communications in Computational Physics VL - 5 SP - 765 EP - 804 PY - 2006 DA - 2006/01 SN - 1 DO - http://dor.org/ UR - https://global-sci.org/intro/article_detail/cicp/7979.html KW - Multi-valued solution KW - level set method KW - high frequency wave propagation. AB -

We review the level set methods for computing multi-valued solutions to a class of nonlinear first order partial differential equations, including Hamilton-Jacobi equations, quasi-linear hyperbolic equations, and conservative transport equations with multi-valued transport speeds. The multivalued solutions are embedded as the zeros of a set of scalar functions that solve the initial value problems of a time dependent partial differential equation in an augmented space. We discuss the essential ideas behind the techniques, the coupling of these techniques to the projection of the interaction of zero level sets and a collection of applications including the computation of the semiclassical limit for Schrödinger equations and the high frequency geometrical optics limits of linear wave equations. 

H. Liu, S. Osher & R. Tsai. (2020). Review Article: Multi-Valued Solution and Level Set Methods in Computational High Frequency Wave Propagation. Communications in Computational Physics. 1 (5). 765-804. doi:
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