New high order multiderivative explicit four-step methods with vanished phase-lag and its derivatives for the approximate solution of the Schrödinger equation. Part I: Construction and theoretical analysis

A new high order implicit four-step methods with vanished phase-lag and some of its derivatives for the numerical solution of the radial Schr¨odinger equation

“…But the developed method in [46] is multiderivative method and hence it implies that more CPU time than the new method presented in this paper for the numerical computations. So, for the numerical solution of the time-independent radial Schrödinger equation the new obtained four-step method with vanished phase-lag and its derivatives is the most accurate ones, especially for large values of |G| = |V c − E|.…”

“…Hence for the classical four-step methods, the error increases as the fifth power of G. For the zero phase-lag multiderivative method presented in [41], the error increases as the fifth power of G. For the implicit P-stable multiderivative method presented in [46], the error increases as the sixth power of G. For the trigonometrically-fitted multiderivative method, first class, produced by Shokri [41], the error increases as the fourth power of G. For the trigonometricallyfitted multiderivative method, second class, produced by Shokri [41], the error increases as the fourth power of G. For the multiderivative method with vanished phase-lag and its first derivative produced by Simos [46], method NMI, the error increases as the third power of G. For the multiderivative method with vanished phase-lag and its first derivative 8 produced by Simos [46], method NMII, the error increases as the third power of G. For the multiderivative method with vanished phase-lag and its first derivative produced by Simos [46], method NMIII, the error increases as the second power of G. For the new four-step method with vanished phase-lag and its first, second and third derivatives obtained in this paper, the error increases as the second power of G with little larger coefficients than the method developed in [46]. But the developed method in [46] is multiderivative method and hence it implies that more CPU time than the new method presented in this paper for the numerical computations.…”

“…An eighth algebraic order multiderivative method with vanished phase-lag and its first and second derivatives (method (9) with coefficients given by (22) in [46]) which is indicated as NMII LT E NMIII = − 614h 10 38102400…”

Exact solution of the Schrödinger equation with a Lennard–Jones potential

“…We have shown the applicability of our method for obtaining global solutions of the Schrödinger equation in the case of bound states in a (12,6) Lennard-Jones potential. The method can be similarly applied to any other Lennard-Jones-type potential, whatever exponents in the attractive and repulsive terms.…”

“…Numerical methods have been developed, among others, by Simos and collaborators [8,9,10,11]. An extensive bibliography concerning those methods can be found in Section 2 of a recent paper [12]. Except for a few familiar potentials, for which the differential equation can be solved exactly [13], those methods provide only with approximate values of the energies and wave functions.…”