Volume 1, Issue 5
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 Schr¨odinger equations and the high frequency geometrical optics limits of linear wave equations. 

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@Article{CiCP-1-765, author = {H. Liu, S. Osher and R. Tsai}, title = {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 Schr¨odinger 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 - Multi-valued Solution and Level Set Methods in Computational High Frequency Wave Propagation AU - H. Liu, S. Osher & R. Tsai 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/cicp/7979.html KW - 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 Schr¨odinger equations and the high frequency geometrical optics limits of linear wave equations. 

H. Liu, S. Osher & R. Tsai. (1970). 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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