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Advances in Numerical Analysis
Volume 2012 (2012), Article ID 913429, 24 pages
Convergence of an Eighth-Order Compact Difference Scheme for the Nonlinear Schrödinger Equation
School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
Received 24 May 2012; Accepted 7 August 2012
Academic Editor: Ting-Zhu Huang
Copyright © 2012 Tingchun Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- D. J. Griffiths, Introduction to Quantum Mechanics, Prentice-Hall, Englewood Cliffs, NJ, USA, 1995.
- C. R. Menyuk, “Stability of solitons in birefringent optical fibers,” Journal of the Optical Society of America B, vol. 5, pp. 392–402, 1998.
- M. Wadati, T. Iizuka, and M. Hisakado, “A coupled nonlinear Schrödinger equation and optical solitons,” Journal of the Physical Society of Japan, vol. 61, no. 7, pp. 2241–2245, 1992.
- G. D. Akrivis, “Finite difference discretization of the cubic Schrödinger equation,” IMA Journal of Numerical Analysis, vol. 13, no. 1, pp. 115–124, 1993.
- T. F. Chan and L. J. Shen, “Stability analysis of difference schemes for variable coefficient Schrödinger type equations,” SIAM Journal on Numerical Analysis, vol. 24, no. 2, pp. 336–349, 1987.
- Q. Chang, E. Jia, and W. Sun, “Difference schemes for solving the generalized nonlinear Schrödinger equation,” Journal of Computational Physics, vol. 148, no. 2, pp. 397–415, 1999.
- W. Z. Dai, “An unconditionally stable three-level explicit difference scheme for the Schrödinger equation with a variable coefficient,” SIAM Journal on Numerical Analysis, vol. 29, no. 1, pp. 174–181, 1992.
- M. Dehghan and A. Taleei, “A compact split-step finite difference method for solving the nonlinear Schrödinger equations with constant and variable coefficients,” Computer Physics Communications, vol. 181, no. 1, pp. 43–51, 2010.
- F. Ivanauskas and M. Radžiūnas, “On convergence and stability of the explicit difference method for solution of nonlinear Schrödinger equations,” SIAM Journal on Numerical Analysis, vol. 36, no. 5, pp. 1466–1481, 1999.
- P. L. Nash and L. Y. Chen, “Efficient finite difference solutions to the time-dependent Schrödinger equation,” Journal of Computational Physics, vol. 130, no. 2, pp. 266–268, 1997.
- J. M. Sanz-Serna, “Methods for the numerical solution of the nonlinear Schroedinger equation,” Mathematics of Computation, vol. 43, no. 167, pp. 21–27, 1984.
- Z. Z. Sun and X. Wu, “The stability and convergence of a difference scheme for the Schrödinger equation on an infinite domain by using artificial boundary conditions,” Journal of Computational Physics, vol. 214, no. 1, pp. 209–223, 2006.
- L. Wu, “Dufort-Frankel-type methods for linear and nonlinear Schrödinger equations,” SIAM Journal on Numerical Analysis, vol. 33, no. 4, pp. 1526–1533, 1996.
- J. M. Sanz-Serna and J. G. Verwer, “Conservative and nonconservative schemes for the solution of the nonlinear Schrödinger equation,” IMA Journal of Numerical Analysis, vol. 6, no. 1, pp. 25–42, 1986.
- Z. Fei, V. M. Pérez-García, and L. Vázquez, “Numerical simulation of nonlinear Schrödinger systems: a new conservative scheme,” Applied Mathematics and Computation, vol. 71, no. 2-3, pp. 165–177, 1995.
- S. Li and L. Vu-Quoc, “Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation,” SIAM Journal on Numerical Analysis, vol. 32, no. 6, pp. 1839–1875, 1995.
- G. D. Akrivis, V. A. Dougalis, and O. A. Karakashian, “On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation,” Numerische Mathematik, vol. 59, no. 1, pp. 31–53, 1991.
- O. Karakashian, G. D. Akrivis, and V. A. Dougalis, “On optimal order error estimates for the nonlinear Schrödinger equation,” SIAM Journal on Numerical Analysis, vol. 30, no. 2, pp. 377–400, 1993.
- Y. Tourigny, “Some pointwise estimates for the finite element solution of a radial nonlinear Schrödinger equation on a class of nonuniform grids,” Numerical Methods for Partial Differential Equations, vol. 10, no. 6, pp. 757–769, 1994.
- W. Bao and D. Jaksch, “An explicit unconditionally stable numerical method for solving damped nonlinear Schrödinger equations with a focusing nonlinearity,” SIAM Journal on Numerical Analysis, vol. 41, no. 4, pp. 1406–1426, 2003.
- T. Iitaka, “Solving the time-dependent Schröinger equation numerically,” Physical Review E, vol. 49, no. 5, pp. 4684–4690, 1994.
- D. Kosloff and R. Kosloff, “A Fourier method solution for the time dependent Schröinger equation as a tool in molecular dynamics,” Journal of Computational Physics, vol. 52, pp. 35–53, 1983.
- B. Li, G. Fairweather, and B. Bialecki, “Discrete-time orthogonal spline collocation methods for Schrödinger equations in two space variables,” SIAM Journal on Numerical Analysis, vol. 35, no. 2, pp. 453–477, 1998.
- M. P. Robinson and G. Fairweather, “Orthogonal spline collocation methods for Schrödinger-type equations in one space variable,” Numerische Mathematik, vol. 68, no. 3, pp. 355–376, 1994.
- G. Berikelashvili, M. M. Gupta, and M. Mirianashvili, “Convergence of fourth order compact difference schemes for three-dimensional convection-diffusion equations,” SIAM Journal on Numerical Analysis, vol. 45, no. 1, pp. 443–455, 2007.
- G. Cohen, High-Order Numerical Methods for Transient Wave Equations, Springer, New York, NY, USA, 2002.
- A. Gopaul and M. Bhuruth, “Analysis of a fourth-order scheme for a three-dimensional convection-diffusion model problem,” SIAM Journal on Scientific Computing, vol. 28, no. 6, pp. 2075–2094, 2006.
- B. Gustafsson and E. Mossberg, “Time compact high order difference methods for wave propagation,” SIAM Journal on Scientific Computing, vol. 26, no. 1, pp. 259–271, 2004.
- B. Gustafsson and P. Wahlund, “Time compact difference methods for wave propagation in discontinuous media,” SIAM Journal on Scientific Computing, vol. 26, no. 1, pp. 272–293, 2004.
- J. C. Kalita, D. C. Dalal, and A. K. Dass, “A class of higher order compact schemes for the unsteady two-dimensional convection-diffusion equation with variable convection coefficients,” International Journal for Numerical Methods in Fluids, vol. 38, no. 12, pp. 1111–1131, 2002.
- S. K. Lele, “Compact finite difference schemes with spectral-like resolution,” Journal of Computational Physics, vol. 103, no. 1, pp. 16–42, 1992.
- H. L. Liao and Z. Z. Sun, “Maximum norm error bounds of ADI and compact ADI methods for solving parabolic equations,” Numerical Methods for Partial Differential Equations, vol. 26, no. 1, pp. 37–60, 2010.
- H. L. Liao, Z. Z. Sun, and H. S. Shi, “Error estimate of fourth-order compact scheme for linear Schrödinger equations,” SIAM Journal on Numerical Analysis, vol. 47, no. 6, pp. 4381–4401, 2010.
- S. Xie, G. Li, and S. Yi, “Compact finite difference schemes with high accuracy for one-dimensional nonlinear Schrödinger equation,” Computer Methods in Applied Mechanics and Engineering, vol. 198, no. 9-12, pp. 1052–1060, 2009.
- D. W. Zingg, “Comparison of high-accuracy finite-difference methods for linear wave propagation,” SIAM Journal on Scientific Computing, vol. 22, no. 2, pp. 476–502, 2000.
- R. A. Horn and C. R. Johnson, Matrix Analysis, chapter 7, Cambridge University Press, 1985.
- F. E. Browder, “Existence and uniqueness theorems for solutions of nonlinear boundary value problems,” in Application of Nonlinear Partial Differential Equations, R. Finn, Ed., Proceedings of Symposia in Applied Mathematics, pp. 24–49, American Mathematical Society, Providence, RI, USA, 1965.