Abstract
In this paper we present the development of a generator of hybrid explicit methods for the numerical solution of the Schrödinger equation. The methods are of algebraic order ten. The coefficients of the generator are calculated appropriately.
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T.E. Simos, Explicit two-step methods with minimal phase-lag for the numerical integration of special second order initial value problems and their application to the one-dimensional Schrödinger equation, J. Comput. Appl. Math. 39 (1992) 89–94.
J.D. Lambert and I.A. Watson, Symmetric multistep methods for periodic initial value problems, J. Inst. Math. Appl. 18 (1976) 189–202.
L. Bruca and L. Nigro A one-step method for direct integration of structural dynamic equations, Internat. J. Numer. Methods Engrg. 15 (1980) 685–699.
M.M. Chawla and P.S. Rao, A Numerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems, J. Comput. Appl. Math. 11 (1984) 277–281.
M.M. Chawla and P.S. Rao, Numerov-type method with minimal phase-lag for the integration of second order periodic initial-value problem. II. Explicit method, J. Comput. Appl. Math. 15 (1986) 329–337.
M.M. Chawla, P.S. Rao and B. Neta, Two-step fourth-order P-stable methods with phase-lag of order six for y″ = f (t,y), J. Comput. Appl. Math 16 (1986) 233–236.
M.M. Chawla and P.S. Rao, An explicit sixth-order method with phase-lag of order eight for y″ = f (t,y), J. Comput. Appl. Math. 16 (1987) 365–368.
J.P. Coleman, Numerical methods for y″ = f (x, y) via rational approximation for the cosine, IMA J. Numer. Anal. 9 (1989) 145–165.
T.E. Simos and A.D. Raptis, Numerov-type methods with minimal phase-lag for the numerical integration of the one-dimensional Schrödinger equation, Computing 45 (1990) 175–181.
T.E. Simos, New variable-step procedure for the numerical integration of the one-dimensional Schrödinger equation, J. Comput. Phys. 108 (1993) 175–179.
G. Avdelas and T.E. Simos, Block Runge-Kutta methods for periodic initial-value problems, Comput. Math. Appl. 31 (1996) 69–83.
G. Avdelas and T.E. Simos, Embedded methods for the numerical solution of the Schrödinger equation, Comput. Math. Appl. 31 (1996) 85–102.
T.E. Simos, New embedded explicit methods with minimal phase-lag for the numerical integration of the Schrödinger equation, Comput. Chem. 22 (1998) 433–440.
T.E. Simos, Numerical solution of ordinary differential equations with periodical solution, Doctoral Dissertation, National Technical University of Athens (1990).
R.M. Thomas, Phase properties of high order, almost P-stable formulae, BIT 24 (1984) 225–238.
T.E. Simos and G. Tougelidis, A Numerov type method for computing eigenvalues and resonances of the radial Schrödinger equation, Comput. Chem. 20 (1996) 397–401.
L.D. Landau and F.M. Lifshitz, Quantum Mechanics (Pergamon, New York, 1965).
I. Prigogine and S. Rice, eds., New Methods in Computational Quantum Mechanics, Advances in Chemical Physics, Vol. 93 (Wiley, New York, 1997).
T.E. Simos, Eighth order methods for accurate computations for the Schrödinger equation, Comput. Phys. Comm. 105 (1997) 127–138.
T.E. Simos, Explicit eighth order methods for the numerical integration of initial-value problems with periodic or oscillating solutions, Comput. Phys. Comm. 119 (1999) 32–44.
G. Avdelas, A. Konguetsof and T.E. Simos, A generator of hybrid explicit methods for the numerical solution of the Schrödinger equation and related problems, Comput. Phys. Comm. (2000), in press.
T.E. Simos, Eighth order methods with minimal phase-lag for accurate computations for the elastic scattering phase-shift problem, J. Math. Chem. 21 (1997) 359–372.
T.E. Simos, Some embedded modified Runge-Kutta methods for the numerical solution of some specific Schrödinger equations, J. Math. Chem. 24 (1998) 23–37.
T.E. Simos, A family of P-stable exponentially-fitted methods for the numerical solution of the Schrödinger equation, J. Math. Chem. 25 (1999) 65–84.
G. Avdelas and T.E. Simos, Embedded eighth order methods for the numerical solution of the Schrödinger equation, J. Math. Chem. 25 (1999) 327–341.
J. Vigo-Aguiar and T.E. Simos, A family of P-stable eighth algebraic order methods with exponential fitting facilities, J. Math. Chem., to appear.
T.E. Simos and P.S. Williams, On finite difference methods for the solution of the Schrödinger equation, Comput. Chem. 23 (1999) 513–554.
T.E. Simos, in: Atomic Structure Computations in Chemical Modelling: Applications and Theory, ed. A. Hinchliffe, UMIST (The Royal Society of Chemistry, 2000) pp. 38–142.
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Avdelas, G., Konguetsof, A. & Simos, T. A Generator and an Optimized Generator of High-Order Hybrid Explicit Methods for the Numerical Solution of the Schrödinger Equation. Part 1. Development of the Basic Method. Journal of Mathematical Chemistry 29, 281–291 (2001). https://doi.org/10.1023/A:1010947219240
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DOI: https://doi.org/10.1023/A:1010947219240