×

Open boundaries for the nonlinear Schrödinger equation. (English) Zbl 1122.65094

Summary: We present a new algorithm, the time dependent phase space filter (TDPSF), which is used to solve time dependent nonlinear Schrödinger equations (NLS). The algorithm consists of solving the NLS on a box with periodic boundary conditions (by any algorithm). Periodically in time we decompose the solution into a family of coherent states. Coherent states which are outgoing are deleted, while those which are not are kept, reducing the problem to reflected (wrapped) waves. Numerical results are given, and rigorous error estimates are described. The TDPSF is compatible with spectral methods for solving the interior problem. The TDPSF also fails gracefully, in the sense that the algorithm notifies the user when the result is incorrect. We are aware of no other method with this capability.

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs

Software:

FFTW
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bayliss, Alvin; Turkel, Eli, Radiation boundary conditions for wave-like equations, Comm. Pure Appl. Math., 33, 6, 707-725 (1980) · Zbl 0438.35043
[2] Bayliss, Alvin; Turkel, Eli, Outflow boundary conditions for fluid dynamics, SIAM J. Sci. Statist. Comput., 3, 2, 250-259 (1982) · Zbl 0509.76035
[3] Berenger, Jean-Pierre, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114, 2, 185-200 (1994) · Zbl 0814.65129
[4] Boyd, John P., Chebyshev and Fourier Spectral Methods (2001), Dover Publications: Dover Publications Mineola, NY · Zbl 0994.65128
[5] Daubechies, Ingrid, Time-frequency localization operators: a geometric phase space approach, IEEE Trans. Inform. Theory, 34, 4, 605-612 (1988) · Zbl 0672.42007
[6] Daubechies, Ingrid, Ten lectures on wavelets, (CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61 (1992), Society for Industrial and Applied Mathematics (SIAM): Society for Industrial and Applied Mathematics (SIAM) Philadelphia, PA) · Zbl 0776.42018
[7] Daubechies, Ingrid; Grossmann, A., Frames in the Bargmann space of entire functions, Comm. Pure Appl. Math., 41, 2, 151-164 (1988) · Zbl 0632.30049
[8] Daubechies, Ingrid; Grossmann, A.; Meyer, Y., Painless nonorthogonal expansions, J. Math. Phys., 27, 5, 1271-1283 (1986) · Zbl 0608.46014
[9] Engquist, Björn; Majda, Andrew, Absorbing boundary conditions for numerical simulation of waves, Proc. Natl. Acad. Sci. USA, 74, 5, 1765-1766 (1977) · Zbl 0378.76018
[10] Engquist, Bjorn; Majda, Andrew, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp., 31, 139, 629-651 (1977) · Zbl 0367.65051
[11] Engquist, Björn; Majda, Andrew, Radiation boundary conditions for acoustic and elastic wave calculations, Comm. Pure Appl. Math., 32, 3, 314-358 (1979) · Zbl 0387.76070
[12] Frigo, Matteo; Johnson, Steven G., The design and implementation of FFTW3, Proceedings of the IEEE, 93, 2, 216-231 (2005), (Special issue on Program Generation, Optimization, and Platform Adaptation)
[13] Givoli, Dan; Neta, Beny, High-order non-reflecting boundary scheme for time-dependent waves, J. Comput. Phys., 186, 1, 24-46 (2003) · Zbl 1025.65049
[14] Hagstrom, Thomas, Radiation boundary conditions for the numerical simulation of waves, (Acta numerica. Acta numerica, Acta Numer, vol. 8 (1999), Cambridge University Press: Cambridge University Press Cambridge), 47-106 · Zbl 0940.65108
[15] Hagstrom, Thomas, New results on absorbing layers and radiation boundary conditions, (Topics in Computational Wave Propagation. Topics in Computational Wave Propagation, Lecturer Notes on Computer Science Engineering, vol. 31 (2003), Springer: Springer Berlin), 1-42 · Zbl 1059.78040
[16] Lubich, Christian; Schädle, Achim, Fast convolution for nonreflecting boundary conditions, SIAM J. Sci. Comput., 24, 1, 161-182 (2002), (electronic) · Zbl 1013.65113
[17] Neuhauser, Daniel; Baer, Micheal, The time-dependent Schrodinger equation: application of absorbing boundary conditions, J. Chem. Phys., 90, 8 (1989)
[18] Gottlieb, D.; Abarbanel, S.; Hesthaven, J. S., Long time behavior of the perfectly matched layer equations in computational electromagnetics, J. Sci. Comput., 17, 1 (2002) · Zbl 1005.78014
[19] Schädle, Achim, Non-reflecting boundary condition for a Schrödinger-type equation, (Mathematical and Numerical Aspects of Wave Propagation (Santiago de Compostela, 2000) (2000), SIAM: SIAM Philadelphia, PA), 621-625 · Zbl 0959.35143
[20] Schädle, Achim, Non-reflecting boundary conditions for the two-dimensional Schrödinger equation, Wave Motion, 35, 2, 181-188 (2002) · Zbl 1163.74435
[21] A. Soffer, C. Stucchio, Multiscale resolution of shortwave-longwave interactions in time dependent dispersive waves (in preparation).; A. Soffer, C. Stucchio, Multiscale resolution of shortwave-longwave interactions in time dependent dispersive waves (in preparation). · Zbl 1155.65084
[22] A. Soffer, C. Stucchio, Time dependent phase space filters: Nonreflecting boundaries for semilinear Schrodinger equations, preprint, 2006.; A. Soffer, C. Stucchio, Time dependent phase space filters: Nonreflecting boundaries for semilinear Schrodinger equations, preprint, 2006.
[23] Szeftel, Jérémie, Design of absorbing boundary conditions for Schrödinger equations in \(R^d\), SIAM J. Numer. Anal., 42, 4, 1527-1551 (2004), (electronic) · Zbl 1094.35037
[24] Szeftel, Jeremie, Absorbing boundary conditions for nonlinear Schrodinger equations, Numerische Mathematik, 104, 103-127 (2006) · Zbl 1130.35119
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.