Research ArticleOpen AccessGoogle Scholar indexed
A Trapezoidal-Like Integrator for the Numerical Solution of One-Dimensional Time Dependent Schrödinger Equation
Department of Mathematical Science and Information Technology, Federal University, Dutsin-Ma, Nigeria
- 1 Department of Mathematical Science and Information Technology, Federal University, Dutsin-Ma, Nigeria
American Journal of Computational Mathematics·Volume 04 (2014)·Pages 271–279·Published 29 August 2014·DOI10.4236/ajcm.2014.44023
Copy link · social · email
Abstract
In this paper, the one-dimensional time dependent Schr?dinger equation is discretized by the method of lines using a second order finite difference approximation to replace the second order spatial derivative. The evolving system of stiff Ordinary Differential Equation (ODE) in time is solved numerically by an L-stable trapezoidal-like integrator. Results show accuracy of relative maximum error of order 10 ?4 in the interval of consideration. The performance of the method as compared to an existing scheme is considered favorable.
KeywordsSchrödinger’s EquationPartial Differential EquationsMethod of Lines (MOL)Stiff ODETrapezoidal-Like Integrator
- Becerril, R., Guzman, F.S., Rendon-Romero, A. and Valdez-Alvarodo, S. (2008) Solving the Time Dependent Schrodinger Equation Using Finite Difference Method. Revista Mexicana de Fisica E, 54, 120-132.
- Jiménez, S., Iorente, M.I.L., Manch, M.A., Peréz-García, M.V. and Vázquez, L. (2003) A Numerical Scheme for the Simulation of Blow-Up of the Nonlinear Schrodinger Equation. Applied Mathematics and Computation, 134, 271-291. http://dx.doi.org/10.1016/S0096-3003(01)00282-X
- Ramos, J.I. and Villatoro, F.R. (1994) The Nonlinear Schrodinger Equation in the Finite Line. Mathematical Computer Modeling, 20, 31-59. http://dx.doi.org/10.1016/0895-7177(94)90030-2
- Zisowsky, A. and Ehrhardt, M. (2008) Discrete Artificial Boundary Condition for Non-Linear Schrodinger Equations. Mathematical and Computer Modeling, 47, 1264-1283. http://dx.doi.org/10.1016/j.mcm.2007.07.007
- Ismail, A.I.N., Karim, F., Roy, G.D. and Meah, M.A. (2007) Numerical Modeling of Tsunami via Method of Lines. World Academy of Science, Engineering and Technology, 1, 341-349.
- Hoz, F. and Vadillo, F. (2008) An Exponential Time Differencing Method for Nonlinear Schrodinger Equation. Computer Physics Communication, 179, 449-456. http://dx.doi.org/10.1016/j.cpc.2008.04.013
- Thron, C. and Watts, J. (2013) A Signal-Processing Interpretation of Quantum Mechanics. The African Review of Physics, 8, 263-270.
- Schiesser, W.E. (1991) Numerical Method of Lines: Integration of Partial Differential Equation. Academic Press Limited, California.
- Schiesser, W.E. and Griffith, G.W. (2009) A Compendium of Partial Differential Equation Model: Method of Lines Analysis with Matlab. Cambridge Universal Press, Cambridge. http://dx.doi.org/10.1017/CBO9780511576270
- Sa’adu, L., Hashim, M.A., Dasuki, K.A. and Sanugi, B. (2012) A Method of Lines Approach in the Numerical Solution of 1-Dimensional Schrodinger Equation. Applied Physics Research, 4, 88-93. http://dx.doi.org/10.5539/apr.v4n3p88
- Samrout, Y.M. (2009) New Second and Fourth Order Accurate Numerical Schemes for the Cubic Schrodinger Equation. International Journal of Computer Mathematics, 8, 1625-1651.
- Proakis, J. (2000) Digital Communications. 4th Edition, McGraw-Hill Publishing, New York.
- Fatokun, J.O. and Akpan, I.P. (2013) L-Stable Implicit Trapezoidal-Like Integrators for the Solution of Parabolic Partial Differential Equations on Manifolds. African Journal of Mathematics and Computer Science Research, 6, 183-190.