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A numerical method for two-dimensional Schrödinger equation using collocation and radial basis functions. (English) Zbl 1126.65092

Summary: We propose a numerical scheme to solve the two-dimensional (2D) time-dependent Schrödinger equation using collocation points and approximating the solution using multiquadrics and the thin plate splines radial basis function. The scheme works in a similar fashion as finite-difference methods. The results of numerical experiments are presented, and are compared with analytical solutions to confirm the good accuracy of the presented scheme.

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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[1] Arnold, A., Numerically absorbing boundary conditions for quantum evolution equations, VLSI Design, 6, 313-319 (1998)
[2] L’evy, M., Parabolic Equation Methods for Electromagnetic Wave Propagation (2000), IEE
[3] Tappert, F. D., The parabolic approximation method, (Keller, J. B.; Papadakis, J. S., Wave Propagation and Underwater Acoustics. Wave Propagation and Underwater Acoustics, Lecture Notes in Physics, vol. 70 (1977), Springer: Springer Berlin), 224-287
[4] Huang, W.; Xu, C.; Chu, S. T.; Chaudhuri, S. K., The finite-difference vector beam propagation method, J. Lightwave Technol., 10, 3, 295-304 (1992)
[5] Hajj, F. Y., Solution of the Schrodinger equation in two and three dimensions, J. Phys. B At. Mol. Phys., 18, 1-11 (1985)
[6] Ixaru, L. Gr., Operations on oscillatory functions, Comput. Phys. Comm., 105, 1-9 (1997) · Zbl 0930.65150
[7] Gasser, I.; Lin, C. K.; Markowich, P. A., A review of dispersive limits of (non) linear Schrödinger type equations, Taiwanese J. Math., 4, 4, 501-529 (2000) · Zbl 0972.35112
[8] Kalita, J. C.; Chhabra, P.; Kumar, S., A semi-discrete higher order compact scheme for the unsteady two-dimensional Schrödinger equation, J. Comput. Appl. Math., 197, 141-149 (2006) · Zbl 1101.65096
[9] Dehghan, M., Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices, Math. Comput. Simulation, 71, 16-30 (2006) · Zbl 1089.65085
[10] Subaşi, M., On the finite-difference schemes for the numerical solution of two dimensional Schrödinger equation, Numer. Methods Partial Differential Equations, 18, 752-758 (2002) · Zbl 1014.65077
[11] Dehghan, M., Fully explicit finite-difference methods for two-dimensional diffusion with an integral condition, Nonlinear Anal., TMA, 48, 5(A), 637-650 (2002) · Zbl 1003.65094
[12] Dehghan, M., Fully implicit finite differences methods for two-dimensional diffusion with a non-local boundary condition, J. Comput. Appl. Math., 106, 2, 255-269 (1999) · Zbl 0931.65091
[13] Antoine, X.; Besse, C.; Mouysett, V., Numerical schemes for the simulation of the two-dimensional Schrödinger equation using non-reflecting boundary conditions, Math. Comput., 73, 1779-1799 (2004) · Zbl 1053.65072
[14] Liu, H.; Yan, J., A local discontinuous Galerkin method for the Korteweg de Vries equation with boundary effect, J. Comput. Phys., 215, 1, 197-218 (2005) · Zbl 1092.65083
[15] Kansa, E. J., Multiquadrics-A scattered data approximation scheme with applications to computational fluid dynamics-I, Comput. Math. Appl., 19, 127-145 (1990) · Zbl 0692.76003
[16] Kansa, E. J., Multiquadrics-A scattered data approximation scheme with applications to computational fluid dynamics-II, Comput. Math. Appl., 19, 147-161 (1990) · Zbl 0850.76048
[17] Hon, Y. C.; Mao, X. Z., An efficient numerical scheme for Burgers equation, Appl. Math. Comput., 95, 1, 37-50 (1998) · Zbl 0943.65101
[18] Hon, Y. C.; Cheung, K. F.; Mao, X. Z.; Kansa, E. J., Multiquadric solution for shallow water equations, ASCE J. Hydraulic Eng., 125, 5, 524-533 (1999)
[19] Zerroukat, M.; Power, H.; Chen, C. S., A numerical method for heat transfer problem using collocation and radial basis functions, Internat. J. Numer. Meth. Engrg., 42, 1263-1278 (1992) · Zbl 0907.65095
[20] Hon, Y. C.; Mao, X. Z., A radial basis function method for solving options pricing model, Financial Engrg., 8, 1, 31-49 (1999)
[21] Marcozzi, M.; Choi, S.; Chen, C. S., On the use of boundary conditions for variational formulations arising in financial mathematics, Appl. Math. Comput., 124, 197-214 (2001) · Zbl 1047.91033
[22] Carlson, R. E.; Foley, T. A., The parameter R2 in multiquadric interpolation, Comput. Math. Appl., 21, 29-42 (1991) · Zbl 0725.65009
[23] A.E. Tarwater, A parameter study of Hardys multiquadric method for scattered data interpolation, Technical report UCRL-54670, Lawrence Livermore National Laboratory, 1985; A.E. Tarwater, A parameter study of Hardys multiquadric method for scattered data interpolation, Technical report UCRL-54670, Lawrence Livermore National Laboratory, 1985
[24] Dehghan, M., The one-dimensional heat equation subject to a boundary integral specification, Chaos Solitons Fractals, 32, 661-675 (2007) · Zbl 1139.35352
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