×

A dissipative exponentially-fitted method for the numerical solution of the Schrödinger equation and related problems. (English) Zbl 1196.65123

Summary: A dissipative exponentially-fitted method for the numerical integration of the Schrödinger equation and related problems is developed. The method is called dissipative since is a nonsymmetric multistep method. An application to the the resonance problem of the radial Schrödinger equation and to other well known related problems indicates that the new method is more efficient than the corresponding classical dissipative method and other well known methods. Based on the new method and the method of Raptis and Cash a new variable-step method is obtained. The application of the new variable-step method to the coupled differential equations arising from the Schrödinger equation indicates the power of the new approach.

MSC:

65L10 Numerical solution of boundary value problems involving ordinary differential equations
65L12 Finite difference and finite volume methods for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Ixaru, L.Gr.; Micu, M., Topics in theoretical physics, (1978), Central Institute of Physics Bucharest
[2] Ixaru, L.Gr.; Rizea, M., A numerov-like scheme for the numerical solution of the Schrödinger equation in the deep continuum spectrum of energies, Comput. phys. commun., 19, 23-27, (1980)
[3] Landau, L.D.; Lifshitz, F.M., Quantum mechanics, (1965), Pergamon New York · Zbl 0178.57901
[4] Herzberg, G., Spectra of diatomic molecules, (1950), Van Nostrand Toronto
[5] Blatt, J.M., Practical points concerning the solution of the Schrödinger equation, J. comput. phys., 1, 382-396, (1967) · Zbl 0182.49702
[6] Cash, J.R.; Raptis, A.D., A high order method for the numerical solution of the one-dimensional Schrödinger equation, Comput. phys. commun., 33, 299-304, (1984)
[7] Cash, J.R.; Raptis, A.D.; Simos, T.E., A sixth-order exponentially fitted method for the numerical solution of the radial Schrödinger equation, J. comput. phys., 91, 413-423, (1990) · Zbl 0717.65056
[8] Cooley, J.W., An improved eigenvalue corrector formula for solving Schrödinger’s equation for central fields, Math. comp., 15, 363-374, (1961) · Zbl 0122.35902
[9] Killingbeck, J., Shooting methods for the Schrödinger equation, J. phys. A: math. gen., 20, 1411-1417, (1987) · Zbl 0627.65096
[10] Kobeissi, H.; Kobeissi, M., On testing difference equations for the diatomic eigenvalue problem, J. comput. chem., 9, 844-850, (1988)
[11] Kobeissi, H.; Kobeissi, M., A new variable step method for the numerical integration of the one-dimensional Schrödinger equation, J. comput. phys., 77, 501-512, (1988) · Zbl 0654.65064
[12] Kobeissi, H.; Kobeissi, M.; El-Hajj, A., On computing eigenvalues of the Schrödinger equation for symmetrical potentials, J. phys. A: math. gen., 22, 287-295, (1989) · Zbl 0677.34023
[13] Kroes, G.J., The royal road to an energy-conserving predictor-corrector method, Comput. phys. commun., 70, 41-52, (1992)
[14] Simos, T.E., Eighth-order methods for accurate computations for the Schrödinger equation, Comput. phys. commun., 105, 127-138, (1997) · Zbl 0930.65088
[15] Raptis, A.D., On the numerical solution of the Schrödinger equation, Comput. phys. commun., 24, 1-4, (1981)
[16] Raptis, A.D., Two-step methods for the numerical solution of the Schrödinger equation, Computing, 28, 373-378, (1982) · Zbl 0473.65060
[17] Raptis, A.D., Exponentially-fitted solutions of the eigenvalue Schrödinger equation with automatic error control, Comput. phys. commun., 28, 427-431, (1983)
[18] Raptis, A.D., Exponential multistep methods for ordinary differential equations, Bull. Greek math. soc., 25, 113-126, (1984) · Zbl 0592.65046
[19] Raptis, A.D.; Allison, A.C., Exponential-Fitting methods for the numerical solution of the Schrödinger equation, Comput. phys. commun., 14, 1-5, (1978)
[20] Raptis, A.D.; Cash, J.R., Exponential and Bessel Fitting methods for the numerical solution of the Schrödinger equation, Comput. phys. commun., 44, 95-103, (1987) · Zbl 0664.65090
[21] T.E. Simos, Numerical solution of ordinary differential equations with periodical solution. Doctoral Dissertation, National Technical University of Athens, 1990
[22] Simos, T.E., A four-step method for the numerical solution of the Schrödinger equation, J. comput. appl. math., 30, 251-255, (1990) · Zbl 0705.65050
[23] Simos, T.E., Some new four-step exponential-Fitting methods for the numerical solution of the radial Schrödinger equation, IMA J. numer. anal., 11, 347-356, (1991) · Zbl 0728.65067
[24] Simos, T.E., Exponential fitted methods for the numerical integration of the Schrödinger equation, Comput. phys. commun., 71, 32-38, (1992)
[25] Simos, T.E., Error analysis of exponential-fitted methods for the numerical solution of the one-dimensional Schrödinger equation, Phys. lett. A, 177, 345-350, (1993)
[26] Vanden Berghe, G.; Fack, V.; De Meyer, H.E., Numerical methods for solving radial Schrödinger equation, J. comput. appl. math., 29, 391-401, (1989) · Zbl 0694.65033
[27] Simos, T.E.; Williams, P.S., A P-stable hybrid exponentially-fitted method for the numerical solution of the Schrödinger equation, Comput. phys. commun., 131, 109-119, (2000) · Zbl 0982.65080
[28] Simos, T.E.; Tougelidis, G., A numerov-type method for computing eigenvalues and resonances of the radial Schrödinger equation, Comput. chemistry, 20, 397, (1996)
[29] Henrici, P., Discrete variable methods in ordinary differential equations, (1962), John Wiley and Sons New York · Zbl 0112.34901
[30] Lyche, T., Chebyshevian multistep methods for ordinary differential equations, Numer. math., 19, 65-75, (1972) · Zbl 0221.65123
[31] Raptis, A.D., Exponential multistep methods for ordinary differential equations, Bull. Greek math. soc., 25, 113-126, (1984) · Zbl 0592.65046
[32] Simos, T.E.; Williams, P.S., A family of numerov-type exponentially-fitted methods for the numerical integration of the Schrödinger equation, Comput. chemistry, 21, 403-417, (1997)
[33] Thomas, R.M.; Simos, T.E., A family of hybrid exponentially-fitted methods for the numerical solution of the radial Schrödinger equation, J. comput. appl. math., 87, 215-226, (1997) · Zbl 0890.65083
[34] Thomas, R.M.; Simos, T.E.; Mitsou, G.V., A family of numerov-type exponentially-fitted methods for the numerical integration of the Schrödinger equation, J. comput. appl. math., 67, 255-270, (1996) · Zbl 0855.65086
[35] Chawla, M.M.; Rao, P.S., A numerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems. II. explicit method, J. comput. appl. math., 15, 329-337, (1986) · Zbl 0598.65054
[36] Dormand, J.R.; Prince, P.J., Runge – kutta – nyström triples, Comput. math. appl., 13, 937-949, (1987) · Zbl 0633.65061
[37] Dormand, J.R.; El-Mikkawy, M.E.; Prince, P.J., Families of runge – kutta – nyström formulae, IMA J. numer. anal., 7, 423-430, (1987) · Zbl 0627.65085
[38] Tsitouras, Ch., Dissipative high phase-lag order methods, Appl. math. comput., 117, 35-43, (2001) · Zbl 1023.65070
[39] Hairer, E.; Norset, S.P.; Wanner, G., Solving ordinary differential equations. I. nonstiff problems, (1993), Springer Berlin · Zbl 0789.65048
[40] Raptis, A.D.; Cash, J.R., A variable step method for the numerical integration of the one-dimensional Schrödinger equation, Comput. phys. commun., 36, 113-119, (1985) · Zbl 0578.65086
[41] Allison, A.C., The numerical solution of coupled differential equations arising from the Schrödinger equation, J. comput. phys., 6, 378-391, (1970) · Zbl 0209.47004
[42] Berstein, R.B.; Dalgarno, A.; Massey, H.; Percival, I.C., Thermal scattering of atoms by homonuclear diatomic molecules, Proc. royal soc. London ser. A, 274, 427-442, (1963)
[43] Berstein, R.B., Quantum mechanical (phase shift) analysis of differential elastic scattering of molecular beams, J. chem. phys., 33, 795-804, (1960)
[44] Avdelas, G.; Simos, T.E., Embedded methods for the for the numerical solution of the Schrödinger equation, Comput. math. appl., 31, 85-102, (1996) · Zbl 0853.65079
[45] Avdelas, G.; Simos, T.E., Embedded eighth order methods for the for the numerical solution of the Schrödinger equation, J. math. chem., 26, 327-341, (1999) · Zbl 0954.65061
[46] Avdelas, G.; Konguetsof, A.; Simos, T.E., A generator of dissipative methods for the numerical solution of the Schrödinger equation, Comput. phys. commun., (2002) · Zbl 1196.65119
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.