Methods and Applications of Analysis

Volume 21 (2014)

Number 1

Fast Huygens sweeping methods for Schrödinger equations in the semi-classical regime

Pages: 31 – 66

DOI: http://dx.doi.org/10.4310/MAA.2014.v21.n1.a2

Authors

Shingyu Leung (Department of Mathematics, the Hong Kong University of Science and Technology, Hong Kong)

Jianliang Qian (Department of Mathematics, Michigan State University, East Lansing, Mich., U.S.A.)

Susana Serna (Department of Mathematics, University of California at Los Angeles)

Abstract

We propose fast Huygens sweeping methods for Schrödinger equations in the semiclassical regime by incorporating short-time Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) propagators into Huygens’ principle. Even though the WKBJ solution is valid only for a short time period due to the occurrence of caustics, Huygens’ principle allows us to construct the global-in-time semi-classical solution. To improve the computational efficiency, we develop analytic approximation formulas for the short-time WKBJ propagator by using the Taylor expansion in time. These analytic formulas allow us to develop two classes of fast Huygens sweeping methods, among which one is posed in the momentum space, and the other is posed in the position space, and both of these methods are of computational complexity $O(N \log N)$ for each time step, where $N$ is the total number of sampling points in the $d$-dimensional position space. To further speed up these methods, we also incorporate the soft-thresholding sparsification strategy into our new algorithms so that the computational cost can be further reduced. The methodology can also be extended to nonlinear Schrödinger equations. One, two, and three dimensional examples demonstrate the performance of the new algorithms.

Keywords

fast Huygens sweeping method, eikonal equation, WKBJ, convolution, fast Fourier transform, Schrödinger equation

2010 Mathematics Subject Classification

65M60, 65N30

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