scholarly journals A New Trigonometrically Fitted Two-Derivative Runge-Kutta Method for the Numerical Solution of the Schrödinger Equation and Related Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Yanwei Zhang ◽  
Haitao Che ◽  
Yonglei Fang ◽  
Xiong You
Keyword(s):  
Schrödinger Equation ◽  
Numerical Solution ◽  
Schrodinger Equation ◽  
Recent Literature ◽  
New Method ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Radial Schrödinger Equation ◽  
Runge Kutta ◽  
Fifth Order

A new trigonometrically fitted fifth-order two-derivative Runge-Kutta method with variable nodes is developed for the numerical solution of the radial Schrödinger equation and related oscillatory problems. Linear stability and phase properties of the new method are examined. Numerical results are reported to show the robustness and competence of the new method compared with some highly efficient methods in the recent literature.

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.

scholarly journals An Optimized Runge-Kutta Method for the Numerical Solution of the Radial Schrödinger Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Qinghe Ming ◽  
Yanping Yang ◽  
Yonglei Fang
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
High Efficiency ◽  
New Method ◽  
Kutta Method ◽  
Saxon Potential ◽  
First Derivative ◽  
Runge Kutta Method ◽  
Radial Schrödinger Equation ◽  
Runge Kutta

An optimized explicit modified Runge-Kutta (RK) method for the numerical integration of the radial Schrödinger equation is presented in this paper. This method has frequency-depending coefficients with vanishing dispersion, dissipation, and the first derivative of dispersion. Stability and phase analysis of the new method are examined. The numerical results in the integration of the radial Schrödinger equation with the Woods-Saxon potential are reported to show the high efficiency of the new method.

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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