Construction of an optimized explicit Runge-Kutta method with increased phase-lag order for the numerical solution of the Schrödinger equation

2012 ◽  
Author(s):  
A. A. Kosti ◽  
Z. A. Anastassi ◽  
T. E. Simos
Keyword(s):  
Numerical Solution ◽  
Phase Lag ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Runge Kutta

AN EMBEDDED RUNGE–KUTTA METHOD WITH PHASE-LAG OF ORDER INFINITY FOR THE NUMERICAL SOLUTION OF THE SCHRÖDINGER EQUATION

2000 ◽  
Vol 11 (06) ◽  
pp. 1115-1133 ◽  
Author(s):  
T. E. SIMOS
Keyword(s):  
Schrödinger Equation ◽  
Numerical Solution ◽  
Numerical Integration ◽  
Schrodinger Equation ◽  
Phase Lag ◽  
Kutta Method ◽  
New Methods ◽  
Runge Kutta Method ◽  
Radial Schrödinger Equation ◽  
Runge Kutta

An embedded Runge–Kutta method with phase-lag of order infinity for the numerical integration of Schrödinger equation is developed in this paper. The methods of the embedded scheme have algebraic orders five and four. Theoretical and numerical results obtained for radial Schrödinger equation and for coupled differential equations show the efficiency of the new methods.

SOME MODIFIED RUNGE–KUTTA METHODS FOR THE NUMERICAL SOLUTION OF SOME SPECIFIC SCHRÖDINGER EQUATIONS AND RELATED PROBLEMS

1996 ◽  
Vol 11 (26) ◽  
pp. 4731-4744 ◽  
Author(s):  
T. E. SIMOS ◽  
P. S. WILLIAMS
Keyword(s):  
Schrödinger Equation ◽  
Numerical Solution ◽  
Schrodinger Equation ◽  
Phase Lag ◽  
Kutta Method ◽  
Oscillatory Solutions ◽  
New Approach ◽  
Runge Kutta Method ◽  
Oscillating Solutions ◽  
Runge Kutta

Some new modified Runge–Kutta methods with minimal phase lag are developed for the numerical solution of the eigenvalue Schrödinger equation and related problems with oscillating solutions. These methods are based on the very well-known Runge–Kutta method of order 4. For the numerical solution of the eigenvalue Schrödinger equation, we investigate two cases: (i) the specific case in which the potential V(x) is an even function with respect to x; it is assumed, also, that the wave functions tend to zero for x → ±∞; (ii) the general case for the well-known cases of the Morse potential and Woods–Saxon or optical potential. Also, we have applied the new methods to some well-known problems with oscillatory solutions. Numerical and theoretical results show that this new approach is more efficient than the well-known classical fourth order Runge–Kutta method and the Numerov method.

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