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Analysis of High-Order Absorbing Boundary Conditions for the Schrödinger Equation

Published online by Cambridge University Press:  20 August 2015

Jiwei Zhang ,
Zhizhong Sun ,
Xiaonan Wu  and
Desheng Wang
Jiwei Zhang*
Affiliation:
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
Zhizhong Sun*
Affiliation:
Department of Mathematics, Southeast University, Nanjing 210096, China
Xiaonan Wu*
Affiliation:
Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong
Desheng Wang*
Affiliation:
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
*
Email:[email protected]
Email:[email protected]
Corresponding author.Email:[email protected]
Email:[email protected]
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Abstract

The paper is concerned with the numerical solution of Schrödinger equations on an unbounded spatial domain. High-order absorbing boundary conditions for one-dimensional domain are derived, and the stability of the reduced initial boundary value problem in the computational interval is proved by energy estimate. Then a second order finite difference scheme is proposed, and the convergence of the scheme is established as well. Finally, numerical examples are reported to confirm our error estimates of the numerical methods.

Keywords

65M1265M0665M15Schrödinger equationfinite difference methodhigh-order absorbing boundary conditionconvergence
Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 3 , September 2011 , pp. 742 - 766
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Alonso, I. Mallo and Reguera, N., Weak ill-posedness of spatial discretizations of absorbing boundary conditions for Schrödinger-type equations, SIAM J. Numer. Anal., 40 (2002), 134–158.Google Scholar
[2]Antoine, X., Besse, C. and Klein, P., Absorbing boundary conditions for Schrödinger equation with an exterior repulsive potential, J. Comput. Phys., 228 (2009), 312–335.Google Scholar
[3]Antoine, X., Arnold, A., Besse, C., Ehrhardt, M. and Schädle, A., A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrädinger equations, Commun. Comput. Phys., 4(4) (2008), 729–796.Google Scholar
[4]Antoine, X. and Besse, C., Unconditionally stable discretization schemes of non-reflecting boundary conditions for the one-dimensional Schrädinger equation, J. Comput. Phys., 188 (2003), 157–175.Google Scholar
[5]Arnold, A., Numerically absorbing boundary conditions for quantum evolution equations, VLSI Design 6, (1-4) (1998), 313–319.Google Scholar
[6]Baskakov, V. and Popov, A., Implementation of transparent boundaries for the numerical solution of the Schrädinger equation, Wave. Motion., 14 (1991), 123–128.Google Scholar
[7]Bécache, E., Givoli, D. and Hagstrom, T., High-order absorbing boundary conditions for anisotropic and convective wave equations, J. Comput. Phys., 229(4) (2010), 1099–1129.Google Scholar
[8]Bérenger, J., A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114 (1994), 185–200.Google Scholar
[9]Bruneau, C., Menza, L. Di and Lehner, T., Numerical resolution of some nonlinear Schrädinger-like equations in plasmas, Numer. Meth. PDEs., 15(6) (1999), 672–696.Google Scholar
[10]Menza, L. Di, Absorbing boundary conditions on a hypersurface for the Schrädinger equation in a half space, Appl. Math. Lett., 9 (1996), 55–59.Google Scholar
[11]Di Menza, L., Transparent and absorbing boundary conditions for the Schrädinger equation in a bounded domain, Numer. Funct. Anal. Opt., 18 (1997), 759–775.Google Scholar
[12]Enguist, B. and Majda, A., Absorbing boundary conditions for the numerical simulation of waves, Math. Comput., 31 (1977), 629–651.Google Scholar
[13]Ehrhardt, M., Discrete transparent boundary conditions for Schrädinger-type equations for non-compactly supported initial data, Appl. Numer. Math., 585(5) (2008), 660–673.Google Scholar
[14]Fevens, T. and Jiang, H., Absorbing boundary conditions for the Schrädinger equation, SIAM J. Sci. Comput., 21(1) (1999), 255–282.Google Scholar
[15]Givoli, D., Numerical Methods for Problems in Infinite Domains, Elsevier, Amsterdam, 1992.Google Scholar
[16]Givoli, D., High-order local non-reflecting boundary conditions: a review, Wave. Motion., 39 (2004), 319–326.Google Scholar
[17]Hagstrom, T. and Warburton, T., A new auxiliary variable formulation of high-order local radiation boundary conditions: corner compatibility conditions and extensions to first order systems, Wave. Motion., 39 (2004), 327–338.Google Scholar
[18]Hagstrom, T., Castro, M. de, Givoli, D. and Tsemach, D., Local high order absorbing boundary conditions for time-dependent waves in guides, J. Comput. Acoust., 15 (2007), 1–22.Google Scholar
[19]Hagstrom, T., Mar-Or, A. and Givoli, D., High-order local absorbing conditions for the wave equation: extensions and improvements, J. Comput. Phys., 227 (2008), 3322–3357.Google Scholar
[20]Hagstrom, T., Radiation boundary conditions for the numerical simulation of waves, Acta. Numer., 8 (1999), 47–106.Google Scholar
[21]Halpern, L. and Rauch, J., Absorbing boundary conditions for diffusion equations, Numer. Math., 71 (1995), 185–224.CrossRefGoogle Scholar
[22]Han, H. and Wu, X., Approximation of infinite boundary condition and its applications to finite element methods, J. Comput. Math., 3 (1985), 179–192.Google Scholar
[23]Han, H. and Wu, X., The Artificial Boundary Method, Tsinghua University Press, 2008.Google Scholar
[24]Han, H., The artificial boundary method-numerical solutions of partial differential equations in unbounded domains, in: Li, T., Zhang, P. (Eds.), Frontiers and Prospects of Contemporary Applied Mathematics, Higher Education Press and World Scientific, (2006), 33–66.Google Scholar
[25]Han, H. and Huang, Z., Exact and approximating boundary conditions for the parabolic problems on unbounded domains, Comput. Math. Appl., 44 (2002), 655–666.Google Scholar
[26]Han, H. and Huang, Z., Exact artificial boundary conditions for the Schrädinger equation in R2, Commun. Math. Sci., 2(1) (2004), 79–94.Google Scholar
[27]Han, H., Jin, J. and Wu, X., A finite-differencemethod for the one-dimensional time-dependent Schrädinger equation on unbounded domain, Comput. Math. Appl., 50 (2005), 1345–1362.Google Scholar
[28]Higdon, R., Absorbing boundary conditions for difference approximations to the multidimensional wave equation, Math. Comput., 47 (1986), 437–459.Google Scholar
[29]Jin, J. and Wu, X., Analysis of finite element method for one-dimensional time-dependent Schrädinger equation on unbounded domain, J. Comput. Appl. Math., 220 (2008), 240–256.Google Scholar
[30]Kuska, J., Absorbing boundary conditions for the Schrädinger equation on finite intervals, Phys. Rev. B., 46(8) (1992), 5000–5003.Google Scholar
[31]Lindmann, E., Free space boundary conditions for the time dependent wave equation, J. Comput. Phys., 18 (1985), 16–78.Google Scholar
[32]Papadakis, J. S., Impedance formulation of the bottom boundary condition for the parabolic equation model in underwater acoustics, NORDA Parabolic Equation Workshop, NORDA Tech. Note 143, 1982.Google Scholar
[33]Shibata, T., Absorbing boundary conditions for the finite difference time domain calculation of the one dimensional Schrädinger equation, Phys. Rev. B., 43 (1991), 6760–6763.CrossRefGoogle ScholarPubMed
[34]Sun, Z. Z. and Wu, X., The stability and convergence of a difference scheme for the Schrädinger equation on an infinite domain by using artificial boundary conditions, J. Comput. Phys., 214 (2006), 209–223.Google Scholar
[35]Sun, Z. Z., The stability and convergence of an explicit difference scheme for the Schrädinger equation on an infinite domain by using artificial boundary conditions, J. Comput. Phys., 219 (2006), 879–898.Google Scholar
[36]Szeftel, J., Design of absorbing boundary conditions for Schrädinger equations in Rd, SIAM J. Numer. Anal., 42(4) (2004), 1527–1551.Google Scholar
[37]Szeftel, J., Absorbing boundary conditions for one-dimensional nonlinear Schrädinger equations, Numer. Math., 104 (2006), 103–127.Google Scholar
[38]Tsynkov, S., Numerical solution of problemson unbounded domains, a review, Appl. Numer. Math., 27 (1998), 465–532.CrossRefGoogle Scholar
[39]Xu, Z., Han, H. and Wu, X., Adaptive absorbing boundary conditions for Schrädinger-type equations: Application to nonlinear and multi-dimensional problems, J. Comput. Phys., 225 (2007), 1577–1589.Google Scholar
[40]Yu, D., Approximation of boundary conditions at infinity for a harmonic equation, J. Comput. Math., 3 (1985), 219–227.Google Scholar
[41]Yu, D., Mathematical Theory of the Natural Boundary Element Method, Science Press, 1993.Google Scholar
[42]Zhang, J., Xu, Z. and Wu, X., Unified approach to split absorbing boundary conditions for nonlinear Schrädinger equations, Phys. Rev. E., 78 (2008), 026709.Google Scholar
[43]Zhang, J., Xu, Z. and Wu, X., Unified approach to split absorbing boundary conditions for nonlinear Schrädinger equations: two dimensional case, Phys. Rev. E., 79 (2009), 046711.Google Scholar
[44]Zhang, J., Local Absorbing Boundary Conditions for Some Nonlinear PDEs on Unbounded Domains, PhD thesis, 2009.Google Scholar
[45]Zheng, C., A perfectly matched layer approachto the nonlinear Schrädinger wave equations, J. Comput. Phys., 227 (2007), 537–556.Google Scholar
[46]Zheng, C., An exact absorbing boundary condition for the Schrädinger equation with sinusoidal potentials at infinity, Commun. Comput. Phys., 3 (2008), 641–658.Google Scholar