We introduce a level set method for computational high frequency wave propagation
in dispersive media and consider the application to linear Schrödinger equation with high frequency
initial data. High frequency asymptotics of dispersive equations often lead to the well-known WKB
system where the phase of the plane wave evolves according to a nonlinear Hamilton-Jacobi equation
and the intensity is governed by a linear conservation law. From the Hamilton-Jacobi equation, wave
fronts with multiple phases are constructed by solving a linear Liouville equation of a vector valued
level set function in the phase space. The multi-valued phase itself can be constructed either from an
additional linear hyperbolic equation in phase space or an additional linear homogeneous equation
and component to the level set function in an augmented phase space. This phase is in fact valid
in the entire physical domain, but one of the components of the level set function can be used to
restrict it to a wave front of interest. The use of the level set method in this numerical approach
provides an Eulerian framework that automatically resolves the multi-valued wave fronts and phase
from the superposition of solutions of the equations in phase space.
Publié le : 2003-09-15
Classification:
Level set method,
Schrödinger’s equation,
semiclassical limit,
wave front,
multivalued phases,
35Q55,
65M25
@article{1250880101,
author = {Cheng, Li-Tien and Liu, Hailiang and Osher, Stanley},
title = {Computational high-frequency wave propogation using the level-set method with
applications to the semi-classical limit of the Schr\"odinger equations},
journal = {Commun. Math. Sci.},
volume = {1},
number = {1},
year = {2003},
pages = { 593-621},
language = {en},
url = {http://dml.mathdoc.fr/item/1250880101}
}
Cheng, Li-Tien; Liu, Hailiang; Osher, Stanley. Computational high-frequency wave propogation using the level-set method with
applications to the semi-classical limit of the Schrödinger equations. Commun. Math. Sci., Tome 1 (2003) no. 1, pp. 593-621. http://gdmltest.u-ga.fr/item/1250880101/