Mathematics of Computation

Discrete absorbing boundary conditions for Schrödinger-type equations. Practical implementation

Author(s): Isaías Alonso-Mallo; Nuria Reguera.
Journal: Math. Comp. 73 (2004), 127-142.
MSC (2000): Primary 65M12, 65M20; Secondary 65M99
Posted: June 6, 2003
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Abstract: Recently, some absorbing boundary conditions for Schrödinger-type equations have been studied by Fevens, Jiang and Alonso-Mallo, and Reguera. These conditions make it possible to obtain a very high absorption at the boundary avoiding the nonlocality of transparent boundary conditions. However, the implementations used in the literature, where the boundary condition is chosen in a manual way in accordance with the solution or fixed independently of the solution, are not practical because of the small absorption. In this paper, a new practical adaptive implementation is developed that allows us to obtain automatically a very high absorption.

1.: I. Alonso-Mallo and N. Reguera, Condiciones frontera transparentes y absorbentes para la ecuación de Schrödinger en una dimensión, Proceedings of the 16th Congress on Differential Equations and Applications and 6th Congress on Applied Mathematics, Las Palmas de Gran Canaria, Spain, 1999, 467-473.
2.: I. Alonso-Mallo and N. Reguera, Weak ill-posedness of spatial discretizations of absorbing boundary conditions for Schrödinger-type equations, SIAM J. Numer. Anal., 40, (2002), 134-158.
3.: I. Alonso-Mallo and N. Reguera, Discrete absorbing boundary conditions for Schrödinger-type equations. Construction and error analysis, to appear in SIAM J. Numer. Anal.
4.: B. Alpert, L. Greengard and T. Hagstrom, Rapid evaluation of nonreflecting boundary kernels for time-domain wave propagation, SIAM J. Numer. Anal., 37, (2000), 1138-1164. MR 2002c:65037
5.: V. A. Baskakov and A. V. Popov, Implementation of transparent boundaries for numerical solution of the Schrödinger equation, Wave Motion, 14, (1991), 123-128. MR 92g:78001
6.: L. Di Menza, Transparent and absorbing boundary conditions for the Schrödinger equation in a bounded domain, Numer. Funct. Anal. and Optimiz., 18, (1997), 759-775. MR 98h:65036
7.: E. Dubach, Artificial boundary conditions for diffusion equations: Numerical study, J. Comp. App. Math., 70, (1996), 127-144. MR 97c:65211
8.: T. Fevens and H. Jiang, Absorbing boundary conditions for the Schrödinger equation, SIAM J. Sci. Comput., 21, (1999), 255-282. MR 2000h:65144
9.: L. Halpern, Absorbing boundary conditions for the discretization schemes of the one-dimensional Wave equation, Math. Comput., 38, (1982), 415-429. MR 83d:65245
10.: L. Halpern and J. Rauch, Absorbing boundary conditions for diffusion equations, Numer. Math., 71, (1995), 185-224. MR 96h:65152
11.: L. Halpern and L.N. Trefethen, Wide-angle one-way equations, J. Acoust. Soc. Am., 84, (1988), 1397-1404. MR 89j:76076
12.: R.L. Higdon, Radiation boundary conditions for dispersive waves, SIAM J. Numer. Anal. 31, (1994), 64-100. MR 94k:35178
13.: W. Huang, C. Xu, S. Chu and S. Chaudhuri, The Finite-Difference Vector Beam Propagation Method: Analysis and Assesment, J. Lightwave Techn., 10, (1992), 295-305.
14.: C. Lubich and A. Schädle, Fast convolution for non-reflecting boundary conditions, SIAM J. Sci. Comput., 21, (2002), 161-182.
15.: S. L. Marple Jr., Digital Spectral Analysis with applications, Prentice Hall, Englewood Cliffs, 1987. MR 89c:93001
16.: A. Messiah, Quantum Mechanichs, Vol.1, trans. G. Temmer, Interscience, New York, 1961. MR 23:B2826
17.: N. Reguera, Stability of a class of matrices with applications to absorbing boundary conditions for Schrödinger-type equations, to appear in Appl. Math. Lett.
18.: N. Reguera, Analysis of a third order absorbing condition for the Schrödinger equation discretized in space, to appear in Appl. Math. Lett.
19.: F. Schmidt, An adaptive approach to the numerical solution of Fresnel's wave equation, J. Lightwave Technol., 11, (1993), 1425-1434.
20.: F. Schmidt and D. Yevick, Discrete transparent boundary conditions for Schrödinger-type equations, J. Comp. Phys., 134, (1997), 96-107. MR 98e:81028

Retrieve articles in Mathematics of Computation with MSC (2000): 65M12, 65M20, 65M99

Isaías Alonso-Mallo
Affiliation: Departamento de Matemática Aplicada y Computación, Universidad de Valladolid, Valladolid, Spain
Email: isaias@mac.cie.uva.es

Nuria Reguera
Affiliation: Departamento de Matemáticas y Computación, Universidad de Burgos, Burgos, Spain
Email: nreguera@ubu.es

DOI: 10.1090/S0025-5718-03-01548-5
PII: S 0025-5718(03)01548-5
Keywords: Schr\"odinger equation, transparent boundary conditions, absorbing boundary conditions
Received by editor(s): November 20, 2001
Received by editor(s) in revised form: May 7, 2002
Posted: June 6, 2003
Additional Notes: The authors have obtained financial support from MCYT BFM 2001-2013 and JCYL VA025/01
Copyright of article: Copyright 2003, American Mathematical Society

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