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A family of high-order multistep methods with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation. (English) Zbl 1236.65083
Summary: Many simulation algorithms (chemical reaction systems, differential systems arising from the modelling of transient behaviour in the process industries etc.) contain the numerical solution of systems of differential equations. For the efficient solution of the above mentioned problems, linear multistep methods or Runge-Kutta single-step methods are used. For the simulation of chemical procedures the radial Schrödinger equation is used frequently. In the present paper we will study a class of linear multistep methods. More specifically, the purpose of this paper is to develop an efficient algorithm for the approximate solution of the radial Schrödinger equation and related problems. This algorithm belongs in the category of the multistep methods. In order to produce an efficient multistep method the phase-lag property and its derivatives are used. Hence the main result of this paper is the development of an efficient multistep method for the numerical solution of systems of ordinary differential equations with oscillating or periodical solutions. The reason of their efficiency, as the analysis proved, is that the phase-lag and its derivatives are eliminated. Another reason of the efficiency of the new obtained methods is that they have high algebraic order.
MSC:
65L06Multistep, Runge-Kutta, and extrapolation methods
81Q05Closed and approximate solutions to quantum-mechanical equations
65L20Stability and convergence of numerical methods for ODE
81-08Computational methods (quantum theory)
References:
[1]Ixaru, L. Gr.; Micu, M.: Topics in theoretical physics, (1978)
[2]Landau, L. D.; Lifshitz, F. M.: Quantum mechanics, (1965) · Zbl 0178.57901
[3], Advances in chemical physics 93 (1997)
[4]Herzberg, G.: Spectra of diatomic molecules, (1950)
[5]Kosti, A. A.; Anastassi, Z. A.; Simos, T. E.: Construction of an optimized explicit Runge–Kutta–Nyström method for the numerical solution of oscillatory initial value problems, Computers mathematics with applications 61, No. 11, 3381-3390 (2011) · Zbl 1222.65066 · doi:10.1016/j.camwa.2011.04.046
[6]Kalogiratou, Z.; Monovasilis, Th.; Simos, T. E.: New modified Runge–Kutta–Nyström methods for the numerical integration of the Schrödinger equation, Computers mathematics with applications 60, No. 6, 1639-1647 (2010) · Zbl 1202.65092 · doi:10.1016/j.camwa.2010.06.046
[7]Konguetsof, A.; Simos, T. E.: An exponentially-fitted and trigonometrically-fitted method for the numerical solution of periodic initial-value problems, Computers mathematics with applications 45, No. 1–3, 547-554 (2003) · Zbl 1035.65071 · doi:10.1016/S0898-1221(03)80036-6
[8]Simos, T. E.: Bessel and Neumann fitted methods for the numerical solution of the Schrödinger equation, Computers mathematics with applications 42, No. 6–7, 833-847 (2001)
[9]Simos, T. E.: A new hybrid embedded variable-step procedure for the numerical integration of the Schrödinger equation, Computers mathematics with applications 36, No. 2, 51-63 (1998) · Zbl 0932.65082 · doi:10.1016/S0898-1221(98)00116-3
[10]Simos, T. E.: An extended numerov-type method for the numerical solution of the Schrödinger equation, Computers mathematics with applications 33, No. 10, 67-78 (1997) · Zbl 0887.65091 · doi:10.1016/S0898-1221(97)00077-1
[11]Papakaliatakis, G.; Simos, T. E.: A new method for the numerical solution of fourth-order bvp’s with oscillating solutions, Computers mathematics with applications 32, No. 10, 1-6 (1996) · Zbl 0874.65058 · doi:10.1016/S0898-1221(96)00181-2
[12]Avdelas, G.; Simos, T. E.: Block Runge–Kutta methods for periodic initial-value problems, Computers mathematics with applications 31, No. 1, 69-83 (1996) · Zbl 0853.65078 · doi:10.1016/0898-1221(95)00183-Y
[13]Avdelas, G.; Simos, T. E.: Embedded methods for the numerical solution of the Schrödinger equation, Computers mathematics with applications 31, No. 2, 85-102 (1996)
[14]Simos, T. E.; Mousadis, G.: A 2-step method for the numerical solution of the radial Schrödinger equation, Computers mathematics with applications 29, No. 7, 31-37 (1995)
[15]Simos, T. E.: Runge–Kutta–Nyström interpolants for the numerical-integration of special 2nd-order periodic initial-value problems, Computers mathematics with applications 26, No. 12, 7-15 (1993)
[16]Simos, T. E.: Runge–Kutta interpolants with minimal phase-lag, Computers mathematics with applications 26, No. 8, 43-49 (1993) · Zbl 0791.65054 · doi:10.1016/0898-1221(93)90330-X
[17]Simos, T. E.: A Runge–Kutta fehlberg method with phase-lag of order infinity for initial-value problems with oscillating solution, Computers mathematics with applications 25, No. 6, 95-101 (1993) · Zbl 0777.65046 · doi:10.1016/0898-1221(93)90303-D
[18]Raptis, A. D.; Allison, A. C.: Exponential–Fitting methods for the numerical solution of the Schrödinger equation, Computer physics communications 14, 1-5 (1978)
[19]Kalogiratou, Zacharoula; Simos, T. E.: A P-stable exponentially-fitted method for the numerical integration of the Schrödinger equation, Applied mathematics and computation 112, 99-112 (2000) · Zbl 1023.65080 · doi:10.1016/S0096-3003(99)00051-X
[20]Raptis, A. D.: Exponentially-fitted solutions of the eigenvalue shrödinger equation with automatic error control, Computer physics communications 28, 427-431 (1983)
[21]Quinlan, G. D.; Tremaine, S.: Symmetric multistep methods for the numerical integration of planetary orbits, The astronomical journal 100, No. 5, 1694-1700 (1990)
[22]Simos, T. E.; Williams, P. S.: A finite-difference method for the numerical solution of the Schrödinger equation, Journal of computational and applied mathematics 79, No. 2, 189-205 (1997) · Zbl 0877.65054 · doi:10.1016/S0377-0427(96)00156-2
[23]Alolyan, Ibraheem; Simos, T. E.: A family of ten-step methods with vanished phase-lag and its first derivative for the numerical solution of the Schrödinger equation, Journal of mathematical chemistry 49, No. 9, 1843-1888 (2011)
[24]Alolyan, Ibraheem; Simos, T. E.: Multistep methods with vanished phase-lag and its first and second derivatives for the numerical integration of the Schrödinger equation, Journal of mathematical chemistry 48, No. 4, 1092-1143 (2010) · Zbl 1202.81028 · doi:10.1007/s10910-010-9728-9
[25]Van Der Houwen, P. J.; Sommeijer, B. P.: Predictor–corrector methods for periodic second-order initial-value problems, The IMA journal of numerical analysis 7, 407-422 (1987) · Zbl 0631.65074 · doi:10.1093/imanum/7.4.407
[26]Ixaru, L. Gr.; Rizea, M.: Comparison of some four-step methods for the numerical solution of the Schrödinger equation, Computer physics communications 38, No. 3, 329-337 (1985) · Zbl 0679.65053 · doi:10.1016/0010-4655(85)90100-6
[27]Lambert, J. D.; Watson, I. A.: Symmetric multistep methods for periodic initial values problems, Journal of the institute of mathematics and its applications 18, 189-202 (1976) · Zbl 0359.65060 · doi:10.1093/imamat/18.2.189
[28]Ixaru, L. Gr.; Rizea, M.: A numerov-like scheme for the numerical solution of the Schrödinger equation in the deep continuum spectrum of energies, Computer physics communications 19, 23-27 (1980)
[29]Dormand, J. R.; El-Mikkawy, M. E. A.; Prince, P. J.: Families of Runge–Kutta–Nyström formulae, The IMA journal of numerical analysis 7, 235-250 (1987) · Zbl 0624.65059 · doi:10.1093/imanum/7.2.235