×

Discrete transparent boundary conditions for Schrödinger-type equations for non-compactly supported initial data. (English) Zbl 1138.65069

Summary: Transparent boundary conditions (TBCs) are an important tool for the truncation of the computational domain in order to compute solutions on an unbounded domain. In this work we want to show how the standard assumption of ‘compactly supported data’ could be relaxed and derive TBCs for a generalized Schrödinger equation directly for the numerical scheme on the discrete level. With this inhomogeneous TBCs it is not necessary that the initial data lies completely inside the computational region. However, an increased computational effort must be accepted.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)

Software:

OASES
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Akrivis, G. D.; Dougalis, V. A.; Zouraris, G. E., Error estimates for finite difference methods for a wide-angle “parabolic” equation, SIAM J. Numer. Anal., 33, 2488-2509 (1996) · Zbl 0862.65047
[2] Antoine, X.; Besse, C., Unconditionally stable discretization schemes of non-reflecting boundary conditions for the one-dimensional Schrödinger equation, J. Comput. Phys., 188, 157-175 (2003) · Zbl 1037.65097
[3] Arnold, A., Numerically absorbing boundary conditions for quantum evolution equations, VLSI Design, 6, 313-319 (1998)
[4] Arnold, A.; Ehrhardt, M., Discrete transparent boundary conditions for wide angle parabolic equations in underwater acoustics, J. Comput. Phys., 145, 611-638 (1998) · Zbl 0915.76081
[5] Arnold, A.; Ehrhardt, M.; Sofronov, I., Discrete transparent boundary conditions for the Schrödinger equation: Fast calculation, approximation, and stability, Comm. Math. Sci., 1, 501-556 (2003) · Zbl 1085.65513
[6] Bamberger, A.; Engquist, B.; Halpern, L.; Joly, P., Parabolic wave equation approximations in heterogeneous media, SIAM J. Appl. Math., 48, 99-128 (1988) · Zbl 0654.35055
[7] Claerbout, J. F., Fundamentals of Geophysical Data Processing (1976), McGraw-Hill: McGraw-Hill New York · Zbl 0156.39202
[8] Collins, M. D., A higher-order parabolic equation for wave propagation in an ocean overlying an elastic bottom, J. Acous. Soc. Am., 86, 1459-1464 (1989)
[9] Ehrhardt, M.; Arnold, A., Discrete transparent boundary conditions for the Schrödinger equation, Riv. Mat. Univ. Parma, 6, 57-108 (2001) · Zbl 0993.65097
[10] M. Ehrhardt, A. Arnold, Discrete transparent boundary conditions for wide angle parabolic equations: Fast calculation and approximation, in: D.G. Simons (Ed.), Proceedings of the Seventh European Conference on Underwater Acoustics, July 3-8, 2004, Netherlands Organisation for Applied Scientific Research (TNO) and Delft University of Technology, Delft, The Netherlands pp. 9-14; M. Ehrhardt, A. Arnold, Discrete transparent boundary conditions for wide angle parabolic equations: Fast calculation and approximation, in: D.G. Simons (Ed.), Proceedings of the Seventh European Conference on Underwater Acoustics, July 3-8, 2004, Netherlands Organisation for Applied Scientific Research (TNO) and Delft University of Technology, Delft, The Netherlands pp. 9-14
[11] Ehrhardt, M.; Zisowsky, A., Discrete non-local boundary conditions for split-step Padé approximations of the one-way Helmholtz equation, J. Comput. Appl. Math., 200, 471-490 (2007) · Zbl 1115.65120
[12] Givoli, D., Non-reflecting boundary conditions, J. Comput. Phys., 94, 1-29 (1991) · Zbl 0731.65109
[13] Givoli, D., Numerical methods for problems in infinite domains, Studies in Applied Mechanics, vol. 33 (1992), Elsevier: Elsevier Amsterdam · Zbl 0788.76001
[14] Jensen, F. B.; Kuperman, W. A.; Porter, M. B.; Schmidt, H., Computational Ocean Acoustics (1994), AIP Press: AIP Press New York
[15] Lee, D.; McDaniel, S. T., Ocean acoustic propagation by finite difference methods, Comput. Math. Appl., 14, 305-423 (1987) · Zbl 0637.76080
[16] Levy, M. F., Parabolic Equation Models for Electromagnetic Wave Propagation, IEE Electromagnetic Waves Series, vol. 45 (2000), IEE: IEE London · Zbl 0943.78001
[17] M.F. Levy, Fast PE models for mixed environments, in: AGARD Conference on Propagation Assessment in Coastal Environments, Bremerhaven, Germany, 19-22 September, 1994, AGARD—Advisory Group for Aerospace Research & Development, Neuilly-sur-Seine, France, North Atlantic Treaty Organization, pp. 8-1-8-6; M.F. Levy, Fast PE models for mixed environments, in: AGARD Conference on Propagation Assessment in Coastal Environments, Bremerhaven, Germany, 19-22 September, 1994, AGARD—Advisory Group for Aerospace Research & Development, Neuilly-sur-Seine, France, North Atlantic Treaty Organization, pp. 8-1-8-6
[18] Levy, M. F., Transparent boundary conditions for parabolic equation: Solutions of radiowave propagation problems, IEEE Trans. Antennas Propag., 45, 66-72 (1997)
[19] Levy, M. F., Non-local boundary conditions for radiowave propagation, (Cohen, G., Third International Conference on Mathematical and Numerical Aspects of Wave Propagation Phenomena. Third International Conference on Mathematical and Numerical Aspects of Wave Propagation Phenomena, Juan-les-Pins, France, 24-28 April 1995 (1995), SIAM: SIAM Philadelphia, PA), 499-505 · Zbl 0874.35120
[20] Mikhin, D., Exact discrete nonlocal boundary conditions for high-order Padé parabolic equations, J. Acous. Soc. Am., 116, 2864-2875 (2004)
[21] J.S. Papadakis, Impedance formulation of the bottom boundary condition for the parabolic equation model in underwater acoustics, NORDA Parabolic Equation Workshop, NORDA Tech. Note 143, 1982; J.S. Papadakis, Impedance formulation of the bottom boundary condition for the parabolic equation model in underwater acoustics, NORDA Parabolic Equation Workshop, NORDA Tech. Note 143, 1982
[22] Papadakis, J. S., Impedance bottom boundary conditions for the parabolic-type approximations in underwater acoustics, (Vichnevetsky, R.; Knight, D.; Richter, G., Advances in Computer Methods for Partial Differential Equations VII (1992), IMACS: IMACS New Brunswick, NJ), 585-590
[23] Papadakis, J. S., Exact nonreflecting boundary conditions for parabolic-type approximations in underwater acoustics, J. Comput. Acous., 2, 83-98 (1994)
[24] Popov, A. V.; Zhu, N. Y., Low-order modes in smoothly curved, oversized and lossy waveguides of arbitrary cross section: Parabolic equation approach, AEÜ Int. J. Electr. Commun., 54, 175-182 (2000)
[25] Schmidt, F.; Friese, T.; Yevick, D., Transparent boundary conditions for split-step Padé approximations of the one-way Helmholtz equation, J. Comput. Phys., 170, 696-719 (2001) · Zbl 0988.65103
[26] Tappert, F. D., The parabolic approximation method, (Keller, J. B.; Papadakis, J. S., Wave Propagation and Underwater Acoustics. Wave Propagation and Underwater Acoustics, Lecture Notes in Physics, vol. 70 (1977), Springer: Springer New York), 224-287 · Zbl 0399.76079
[27] Thomson, D. J.; Mayfield, M. E., An exact radiation condition for use with the a posteriori PE method, J. Comput. Acous., 2, 113-132 (1994)
[28] Tsynkov, S. V., Numerical solution of problems on unbounded domains. A review, Appl. Numer. Math., 27, 465-532 (1998) · Zbl 0939.76077
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.