Exponentially‐Fitted Runge‐Kutta‐Nyström Methods for the Solution of the Schrödinger Equation

2008 ◽  
Author(s):  
Z. Kalogiratou ◽  
Th. Monovasilis ◽  
T. E. Simos
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nyström Methods ◽  
Exponentially Fitted ◽  
Runge Kutta

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.

EXPONENTIALLY FITTED RUNGE–KUTTA FOURTH ALGEBRAIC ORDER METHODS FOR THE NUMERICAL SOLUTION OF THE SCHRÖDINGER EQUATION AND RELATED PROBLEMS

2000 ◽  
Vol 11 (04) ◽  
pp. 785-807 ◽  
Author(s):  
P. S. WILLIAMS ◽  
T. E. SIMOS
Keyword(s):  
Schrödinger Equation ◽  
Numerical Solution ◽  
Fourth Order ◽  
Numerical Experiments ◽  
Schrodinger Equation ◽  
Variable Step ◽  
Algebraic Order ◽  
Exponentially Fitted ◽  
Runge Kutta ◽  
Step Algorithm

Fourth order exponential and trigonometric fitted Runge–Kutta methods are developed in this paper. They are applied to problems involving the Schrödinger equation and to other related problems. Numerical results show the superiority of these methods over conventional fourth order Runge–Kutta methods. Based on the methods developed in this paper, a variable-step algorithm is proposed. Numerical experiments show the efficiency of the new algorithm.

EXPONENTIALLY FITTED TWO-DERIVATIVE RUNGE–KUTTA METHODS FOR THE SCHRÖDINGER EQUATION

2013 ◽  
Vol 24 (10) ◽  
pp. 1350073 ◽  
Author(s):  
YONGLEI FANG ◽  
XIONG YOU ◽  
QINGHE MING
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
High Efficiency ◽  
New Methods ◽  
Asymptotic Expressions ◽  
Radial Schrödinger Equation ◽  
Algebraic Order ◽  
Exponentially Fitted ◽  
Runge Kutta ◽  
Order Four

Two exponentially fitted two-derivative Runge–Kutta (EFTDRK) methods of algebraic order four are derived. The asymptotic expressions of the local errors for large energies are obtained. The numerical results in the integration of the radial Schrödinger equation with the Woods–Saxon potential show the high efficiency of our new methods compared to some famous optimized codes in the literature.

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