A High-Order Two-Step Phase-Fitted Method for the Numerical Solution of the Schrödinger Equation

In this paper, we will develop a four-stage high algebraic order symmetric two-step method with vanished phase-lag and its first up to the fourth derivative. For the proposed method, we will study the following: the phase-lag analysis of the new method; the development of the new method; the local truncation error analysis which is based on the radial Schrödinger equation; the stability and the interval of periodicity analysis which is based on a scalar test equation with frequency different than the frequency of the scalar test equation used for the phase-lag analysis; the error estimation procedure which is based on the algebraic order; and the numerical results from our numerical tests for the examination of the efficiency of the new obtained method. The numerical tests are based on the numerical solution of the Schrödinger equation.

[1]  An Optimized Symmetric 8-Step Semi-Embedded Predictor-Corrector Method for IVPs with Oscillating Solutions , 2013 .

[2]  J. Lambert,et al.  Symmetric Multistip Methods for Periodic Initial Value Problems , 1976 .

[3]  Richard B. Bernstein,et al.  Quantum Mechanical (Phase Shift) Analysis of Differential Elastic Scattering of Molecular Beams , 1960 .

[4]  J. Lambert.  Numerical Methods for Ordinary Differential Systems: The Initial Value Problem , 1991 .

[5]  J. Dormand,et al.  A family of embedded Runge-Kutta formulae , 1980 .

[6]  Zacharias A. Anastassi,et al.  An optimized Runge-Kutta method for the solution of orbital problems , 2005 .

[7]  A. C. Allison,et al.  Exponential-fitting methods for the numerical solution of the schrodinger equation , 1978 .

[8]  Minjian Liang,et al.  A new four stages symmetric two-step method with vanished phase-lag and its first derivative for the numerical integration of the Schrödinger equation , 2016, Journal of Mathematical Chemistry.

[9]  A. D. Raptis,et al.  A variable step method for the numerical integration of the one-dimensional Schrödinger equation , 1985 .

[10]  J. M. Franco,et al.  High-order P-stable multistep methods , 1990 .

[11]  A. C. Allison,et al.  The numerical solution of coupled differential equations arising from the Schrödinger equation , 1970 .

[12]  T. E. Simos,et al.  A new two stage symmetric two-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the radial Schrödinger equation , 2015, Journal of Mathematical Chemistry.

[13]  A. C. Allison,et al.  The rotational excitation of molecular hydrogen , 1967 .

[14]  Theodore E. Simos,et al.  A parametric symmetric linear four-step method for the efficient integration of the Schrödinger equation and related oscillatory problems , 2012, J. Comput. Appl. Math..

[15]  Theodore E. Simos,et al.  A new approach on the construction of trigonometrically fitted two step hybrid methods , 2016, J. Comput. Appl. Math..

[16]  R. Thomas,et al.  Phase properties of high order, almostP-stable formulae , 1984 .

[17]  T. E. Simos,et al.  A Modified Runge-Kutta-Nystr¨ om Method by using Phase Lag Properties for the Numerical Solution of Orbital Problems , 2013 .

[18]  T. E. Simos,et al.  A new high algebraic order four stages symmetric two-step method with vanished phase-lag and its first and second derivatives for the numerical solution of the Schrödinger equation and related problems , 2016, Journal of Mathematical Chemistry.

[19]  Theodore E. Simos,et al.  Exponentially Fitted Symplectic Runge-Kutta-Nystr om methods , 2013 .

[20]  T. E. Simos,et al.  Exponentially fitted Runge-Kutta methods for the numerical solution of the Schrödinger equation and related problems , 2000 .

[21]  Tom E. Simos,et al.  A finite-difference method for the numerical solution of the Schro¨dinger equation , 1997 .

[22]  M. M. Chawla,et al.  Families of two-step fourth order P-stable methods for second oder differential equations , 1986 .

[23]  T. Simos,et al.  A Runge–Kutta type implicit high algebraic order two-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of coupled differential equations arising from the Schrödinger equation , 2015, Journal of Mathematical Chemistry.

[24]  Alexander Dalgarno,et al.  Thermal scattering of atoms by homonuclear diatomic molecules , 1963, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[25]  A. D. Raptis,et al.  A four-step phase-fitted method for the numerical integration of second order initial-value problems , 1991 .