zbMATH — the first resource for mathematics

The use of compact boundary value method for the solution of two-dimensional Schrödinger equation. (English) Zbl 1159.65081
Summary: A high-order and accurate method is proposed for solving the unsteady two-dimensional Schrödinger equation. We apply a compact finite difference approximation of fourth-order for discretizing the spatial derivatives and a boundary value method of fourth-order for the time integration of the resulting linear system of ordinary differential equations. The proposed method has fourth-order accuracy in both space and time variables. Moreover this method is unconditionally stable due to the favorable stability property of boundary value methods.
The results of numerical experiments are compared with analytical solutions and with those provided by other methods in the literature. These results show that the combination of a compact finite difference approximation of fourth-order and a fourth-order boundary value method gives an efficient algorithm for solving the two dimensional Schrödinger equation.

65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
35Q40 PDEs in connection with quantum mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65L05 Numerical methods for initial value problems
Full Text: DOI
[1] Amodio, P.; Mazzia, F.; Trigiante, D., Stability of some boundary value methods for the solution of initial value problems, Bit, 33, 434-451, (1993) · Zbl 0795.65041
[2] Arnold, A., Numerically absorbing boundary conditions for quantum evolution equations, VLSI design, 6, 313-319, (1998)
[3] Brugnano, L.; Trigiante, D., Solving differential problems by multistep initial and boundary value methods, (1998), Gordon and Beach Science Publishers Amsterdam · Zbl 0934.65074
[4] Brugnano, L.; Trigiante, D., Stability properties of some BVM methods, Appl. numer. math., 13, 291-304, (1993) · Zbl 0805.65076
[5] Brugnano, L.; Trigiante, D., Boundary value methods: the third way between linear multistep and runge – kutta methods, Comput. math. appl., 36, 10-12, 269-284, (1998) · Zbl 0933.65082
[6] 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
[7] Dehghan, M.; Shokri, A., A numerical method for two-dimensional Schrödinger equation using collocation and radial basis functions, Comput. math. appl., 54, 136-146, (2007) · Zbl 1126.65092
[8] M. Dehghan, D. Mirzaei, Numerical solution to the unsteady two-dimensional Schrödinger equation using meshless local boundary integral equation method, Int. J. Numer. Meth. Engng. (in press) · Zbl 1195.81007
[9] Ghelardoni, P.; Marzulli, P., Stability of some boundary value methods for ivps, Appl. numer. math., 18, 141-153, (1995) · Zbl 0834.65080
[10] Hajj, F.Y., Solution of the Schrödinger equation in two and three dimensions, J. phys. B, 18, 1-11, (1985)
[11] 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)
[12] Iavernaro, F.; Mazzia, F., Block-boundary value methods for the solution of ordinary differential equations, SIAM J. sci. comput., 21, 323-339, (1999) · Zbl 0941.65067
[13] Ixaru, L.Gr., Operations on oscillatory functions, Comput. phys. comm., 105, 1-9, (1997) · Zbl 0930.65150
[14] 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
[15] Kim, S., Compact schemes for acoustics in the frequency domain, Math. comput. modelling, 37, 1335-1341, (2003) · Zbl 1053.76048
[16] Kopylov, Y.V.; Popov, A.V.; Vinogradov, A.V., Applications of the parabolic wave equations to X-ray diffraction optics, Optics comm., 118, 619-636, (1995)
[17] Lévy, M., Parabolic equation methods for electromagnetic wave propagations, (2000), IEEE
[18] Shang, J.S., High-order compact difference schemes for time-dependent Maxwell equations, J. comput. phys., 153, 312-333, (1999) · Zbl 0956.78018
[19] W.F. Spotz, High-order compact finite difference schemes for computational mechanics, Ph.D. Thesis, University of Texas at Austin, Austin, TX, 1995
[20] Subasi, 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
[21] Sun, H.; Zhang, J., A high-order compact boundary value method for solving one-dimensional heat equations, Numer. methods partial differential equations, 19, 846-857, (2003) · Zbl 1038.65084
[22] Tappert, F.D., The parabolic approximation method, (), 224-287
[23] Dehghan, M., Parameter determination in a partial differential equation from the overspecified data, Math. comput. modelling, 41, 196-213, (2005) · Zbl 1080.35174
[24] Dehghan, M., A computational study of the one-dimensional parabolic equation subject to nonclassical boundary specifications, Numer. methods partial differential equations, 22, 220-257, (2006) · Zbl 1084.65099
[25] Shakeri, F.; Dehghan, M., Numerical solution of a biological population model using he’s variational iteration method, Comput. math. appl., 54, 1197-1209, (2007) · Zbl 1137.92033
[26] Dehghan, M.; Shakeri, F., Application of he’s variational iteration method for solving the Cauchy reaction – diffusion problem, J. comput. appl. math., 214, 435-446, (2008) · Zbl 1135.65381
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.