Fast Hyigens sweeping methods for Schrodinger equations in the semi-classical regime
Leung, Shingyu (Hong Kong University of Science and Technology. Department of Mathematics)
Quian, Jianliang (Michigan State University. Department of Mathematics)
Serna, Susana (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Date:	2014
Abstract:	We propose fast Huygens sweeping methods for Schrodinger equations in the semi-classical 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 Schrodinger equations. One, two, and three dimensional examples demonstrate the performance of the new algorithms.
Note:	Agraïments: This paper is dedicated to Prof. Stan Osher on the occasion of his 70th birthday. Leung is supported in part by the Hong Kong RGC under Grant GRF603011. Qian is supported by NS.
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Convolution ; Eikonal equation ; Fast Fourier transform ; Fast Huygens sweeping method ; Schrodinger equation ; WKBJ
Published in:	Methods and Applications of Analysis, Vol. 21 Núm. 1 (2014) , p. 31-66, ISSN 1945-0001

Postprint 35 p, 6.3 MB

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