Computational high-frequency wave propogation using the level-set method with applications to the semi-classical limit of the Schrödinger equations

Abstract

We introduce a level set method for computational high frequency wave propagation in dispersive media and consider the application to linear Schrodinger 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.