Runge-Kutta Type Tenth Algebraic Order Method with Vanished Phase-Lag and its First, Second, Third, Fourth and Fifth Derivatives for the Numerical Solution of the Schrodinger Equation

2016 ◽  
Vol 10 (1) ◽  
pp. 143-153 ◽  
Author(s):  
Fei Hui ◽  
T. E. Simos
Keyword(s):  
Schrödinger Equation ◽  
Numerical Solution ◽  
Schrodinger Equation ◽  
Phase Lag ◽  
Order Method ◽  
Algebraic Order ◽  
Runge Kutta

A NEW MODIFIED RUNGE–KUTTA–NYSTRÖM METHOD WITH PHASE-LAG OF ORDER INFINITY FOR THE NUMERICAL SOLUTION OF THE SCHRÖDINGER EQUATION AND RELATED PROBLEMS

2000 ◽  
Vol 11 (06) ◽  
pp. 1195-1208 ◽  
Author(s):  
T. E. SIMOS ◽  
JESUS VIGO AGUIAR
Keyword(s):  
Schrödinger Equation ◽  
Numerical Solution ◽  
Schrodinger Equation ◽  
New Method ◽  
Phase Lag ◽  
New Approach ◽  
Nyström Methods ◽  
Oscillating Solutions ◽  
Algebraic Order ◽  
Runge Kutta

In this paper, a new approach for developing efficient Runge–Kutta–Nyström methods is introduced. This new approach is based on the requirement of annihilation of the phase-lag (i.e., the phase-lag is of order infinity) and on a modification of Runge–Kutta–Nyström methods. Based on this approach, a new modified Runge–Kutta–Nyström fourth algebraic order method is developed for the numerical solution of Schrödinger equation and related problems. The new method has phase-lag of order infinity and extended interval of periodicity. Numerical illustrations on the radial Schrödinger equation and related problems with oscillating solutions indicate that the new method is more efficient than older ones.

A NEW METHODOLOGY FOR THE CONSTRUCTION OF OPTIMIZED RUNGE–KUTTA–NYSTRÖM METHODS

2011 ◽  
Vol 22 (06) ◽  
pp. 623-634 ◽  
Author(s):  
D. F. PAPADOPOULOS ◽  
T. E. SIMOS
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
First Integrals ◽  
New Method ◽  
Phase Lag ◽  
Nyström Methods ◽  
Algebraic Order ◽  
Runge Kutta ◽  
First Time ◽  
Zero Phase

In this paper, a new Runge–Kutta–Nyström method of fourth algebraic order is developed. The new method has zero phase-lag, zero amplification error and zero first integrals of the previous properties. Numerical results indicate that the new method is very efficient for solving numerically the Schrödinger equation. We note that for the first time in the literature we use the requirement of vanishing the first integrals of phase-lag and amplification error in the construction of efficient methods for the numerical solution of the Schrödinger equation.

scholarly journals The Use of Phase Lag and Amplification Error Derivatives for the Construction of a Modified Runge-Kutta-Nyström Method

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
D. F. Papadopoulos ◽  
T. E. Simos
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Numerical Results ◽  
Phase Lag ◽  
Nyström Method ◽  
Nystrom Method ◽  
Modified Method ◽  
Algebraic Order ◽  
Runge Kutta

A new modified Runge-Kutta-Nyström method of fourth algebraic order is developed. The new modified RKN method is based on the fitting of the coefficients, due to the nullification not only of the phase lag and of the amplification error, but also of their derivatives. Numerical results indicate that the new modified method is much more efficient than other methods derived for solving numerically the Schrödinger equation.

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