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Semi-Eulerian and High Order Gaussian Beam Methods for the Schrödinger Equation in the Semiclassical Regime

Published online by Cambridge University Press:  20 August 2015

Shi Jin ,
Hao Wu  and
Xu Yang
Shi Jin*
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA
Hao Wu*
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing 10084, China
Xu Yang*
Affiliation:
Program in Applied and Computational Mathematics, Princeton University, NJ 08544, USA
*
Corresponding author.Email:jin@math.wisc.edu
Email:hwu@tsinghua.edu.cn
Email:xuyang@math.princeton.edu
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Abstract

A novel Eulerian Gaussian beam method was developed in [8] to compute the Schrödinger equation efficiently in the semiclassical regime. In this paper, we introduce an efficient semi-Eulerian implementation of this method. The new algorithm inherits the essence of the Eulerian Gaussian beam method where the Hessian is computed through the derivatives of the complexified level set functions instead of solving the dynamic ray tracing equation. The difference lies in that, we solve the ray tracing equations to determine the centers of the beams and then compute quantities of interests only around these centers. This yields effectively a local level set implementation, and the beam summation can be carried out on the initial physical space instead of the phase plane. As a consequence, it reduces the computational cost and also avoids the delicate issue of beam summation around the caustics in the Eulerian Gaussian beam method. Moreover, the semi-Eulerian Gaussian beam method can be easily generalized to higher order Gaussian beam methods, which is the topic of the second part of this paper. Several numerical examples are provided to verify the accuracy and efficiency of both the first order and higher order semi-Eulerian methods.

Keywords

81Q2065M99Schrödinger equationsemi-Eulerian Gaussian beam methodhigh order methods
Type
Research Article
Information
Communications in Computational Physics , Volume 9 , Issue 3 , March 2011 , pp. 668 - 687
Copyright
Copyright © Global Science Press Limited 2011

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