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.

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 A New Optimized Runge-Kutta Pair for the Numerical Solution of the Radial Schrödinger Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Yonglei Fang ◽  
Qinghong Li ◽  
Qinghe Ming ◽  
Kaimin Wang
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Phase Lag ◽  
Order Method ◽  
First Derivative ◽  
New Methods ◽  
Radial Schrödinger Equation ◽  
Runge Kutta ◽  
Fourth Order Method ◽  
Fifth Order

A new embedded pair of explicit modified Runge-Kutta (RK) methods for the numerical integration of the radial Schrödinger equation is presented. The two RK methods in the pair have algebraic orders five and four, respectively. The two methods of the embedded pair are derived by nullifying the phase lag, the first derivative of the phase lag of the fifth-order method, and the phase lag of the fourth-order method. Nu merical experiments show the efficiency and robustness of our new methods compared with some well-known integrators in the literature.

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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