Abstract
In this paper, split-step cubic B-spline collocation (SS3BC) schemes are constructed by combining the split-step approach with the cubic B-spline collocation (3BC) method for the nonlinear Schrödinger (NLS) equation in one, two, and three dimensions with Neumann boundary conditions. Unfortunately, neither of the advantages of the two methods can be maintained for the multi-dimensional problems, if one combines them in the usual manner. For overcoming the difficulty, new medium quantities are introduced in this paper to successfully reduce the multi-dimensional problems into one-dimensional ones, which are essential for the SS3BC methods. Numerical tests are carried out, and the schemes are verified to be convergent with second-order both in time and space. The proposed method is also compared with the split-step finite difference (SSFD) scheme. Finally, the present method is applied to two problems of the Bose-Einstein condensate. The proposed SS3BC method is numerically verified to be effective and feasible.
Similar content being viewed by others
References
Fairweather, G., Meade, D.: A survey of spline collocation methods for the numerical solution of differential equations. In: Diaz, J.C. (ed.) Mathematics for Large Scale Computing, Lecture Notes in Pure and Applid Mathematics, vol. 120. Marcel Dekker, New York (1989)
Daǧ, İ., Saka, B., Irk, D.: Application of cubic B-splines for numerical solution of the RLW equation. Appl. Math. Comput. 159, 373–389 (2004)
Khalifa, A.K., Raslan, K.R., Alzubaidi, H.M.: A collocation method with cubic B-splines for solving the MRLW equation. J. Comput. Appl. Math. 212, 406–418 (2008)
Mittal, R.C., Bhatia, R.: Numerical solution of nonlinear system of Klein-Gordon equations by cubic B-spline collocation method. Inter. J. Comput. Math. 92(10), 2139–2159 (2015)
Pourgholi, R., Saeedi, A.: Applications of cubic B-splines collocation method for solving nonlinear inverse parabolic partial differential equations. Numer. Methods Partial Diff. Eq. 33, 88–104 (2017)
Wang, H.: Numerical studies on the split-step finite difference method for nonlinear Schrödinger equations. Appl. Math. Comput. 170, 17–35 (2005)
Wang, S., Zhang, L.: An efficient split-step compact finite difference method for cubic-quintic complex Ginzburg-Landau equations. Comput. Phys. Commun. 184, 1511–1521 (2013)
Wang, H., Ma, X., Lu, J., et al.: An efficient time-splitting compact finite difference method for Gross-Pitaevskii equation. Appl. Math. Comput. 297, 131–144 (2017)
Gao, Y., Mei, L., Li, R.: A time-splitting Galerkin finite element method for Davey-Stewartson equations. Comput. Phys. Commun. 197, 35–42 (2015)
Bao, W., Shen, J.: A fourth-order time-splitting Laguerre-Hermite pseudospectral method for Bose-Einstein condensates. SIAM J. Sci. Comput. 26(6), 2010–2028 (2005)
Wang, S., Wang, T., Zhang, L.: Numerical computations for N-coupled nonlinear Schrödinger equations by split step spectral methods. Appl. Math. Comput. 222, 438–452 (2013)
Hofstätter, H., Koch, O., Thalhammer, M.: Convergence analysis of high-order time-splitting pseudo-spectral methods for rotational Gross-Pitaevskii equations. Numer. Math. 127, 315–364 (2014)
Wang, S., Zhang, L.: Split-step orthogonal spline collocation methods for nonlinear Schrödinger equations in one, two, and three dimensions. Appl. Math. Comput. 218, 1903–1916 (2011)
Yağmurlu, N.M., Uçar, Y., Çelikkaya, İ. : Operator splitting for numerical solutions of the RLW equation. J. Appl. Anal. Comput. 8(5), 1494–1510 (2018)
Saka, B., Daǧ, İ.: Quartic B-spline collocation method to the numerical solutions of the Burgers’ equation. Chaos Soliton Fract. 32, 1125–1137 (2007)
Saka, B.: Algorithms for numerical solution of the modified equal width wave equation using collocation method. Math. Comput. Model. 45, 1096–1117 (2007)
Botella, O.: On a collocation B-spline method for the solution of the Navier-Stokes equations. Comput. Fluids 31, 397–420 (2002)
Johnson, R.W.: A B-spline collocation method for solving the incompressible Navier-Stokes equations using an hoc method: the boundary residual method. Comput. Fluids 34, 121–149 (2005)
Hidayat, M.I.P., Ariwahjoedi, B., Parman, S.: B-spline collocation with domain decomposition method. J. Phys.: Conf. Ser. 423, 012012 (2013)
Taha, T.R., Ablowitz, M.J.: Analytical and numerical aspects of certain nonlinear Schrödinger equations. II: Numerical. J. Comput. Phys. 55, 203–230 (1984)
Fairweather, G., Khebchareon, M.: Numerical Methods for Schrödinger-Type Problems, Trends in Industrial and Applied Mathematics. Kluwer Academic, Netherlands (2002)
Prenter, P.M.: Splines and Variational Methods. Wiley, New York (1975)
Strang, G.: On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5(3), 506–517 (1968)
Yu, S., Zhao, S., Wei, G.W.: Local spectral time splitting method for first- and second-order partial differential equations. J. Comput. Phys. 206, 727–780 (2005)
Fernandes, R.I.: Efficient orthogonal spline collocation methods for solving linear second order hyperbolic problems on rectangles. Numer. Math. 77, 223–241 (1997)
Bao, W., Jaksch, D., Markowich, P.A.: Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation. J. Comput. Phys. 187, 318–342 (2003)
Funding
This work is supported by the National Natural Science Foundation of China under Grant No. 11701280.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Wang, S., Zhang, L. Split-step cubic B-spline collocation methods for nonlinear Schrödinger equations in one, two, and three dimensions with Neumann boundary conditions. Numer Algor 81, 1531–1546 (2019). https://doi.org/10.1007/s11075-019-00762-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-019-00762-2