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

1999 ◽  
Vol 10 (05) ◽  
pp. 839-851 ◽  
Author(s):  
T. E. SIMOS ◽  
P. S. WILLIAMS
Keyword(s):  
Schrödinger Equation ◽  
Numerical Solution ◽  
Numerical Integration ◽  
Schrodinger Equation ◽  
Numerical Results ◽  
New Methods ◽  
Algebraic Order ◽  
Exponentially Fitted ◽  
Runge Kutta

Exponentially and trigonometrically fitted third algebraic order Runge–Kutta methods for the numerical integration of the Schrödinger equation are developed in this paper. Numerical results obtained for several well known problems show the efficiency of the new methods.

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.

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.

An Accurate Method for the Numerical Solution of the Schrödinger Equation

1997 ◽  
Vol 12 (26) ◽  
pp. 1891-1900 ◽  
Author(s):  
T. E. Simos
Keyword(s):  
Schrödinger Equation ◽  
Numerical Solution ◽  
Numerical Integration ◽  
Schrodinger Equation ◽  
Free Parameter ◽  
Accurate Method ◽  
Numerical Results ◽  
Exponential Functions ◽  
New Methods ◽  
Radial Schrödinger Equation

We present here an accurate method for the numerical integration of the radial Schrödinger equation. The formula considered contains free parameter which are defined in order to integrate exponential functions. Numerical results also indicate that the new methods are much more accurate than other Numerov-type well-known methods.

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.

An Eighth Order Exponentially Fitted Method for the Numerical Solution of the Schrödinger Equation

1998 ◽  
Vol 09 (02) ◽  
pp. 271-288 ◽  
Author(s):  
T. E. Simos
Keyword(s):  
Schrödinger Equation ◽  
Numerical Solution ◽  
Numerical Integration ◽  
Schrodinger Equation ◽  
New Method ◽  
Exponential Functions ◽  
Eighth Order ◽  
Exponentially Fitted ◽  
Exponentially Fitted Method ◽  
Free Parameters

An eighth order exponentially fitted method is developed for the numerical integration of the Schrödinger equation. The formula considered contains certain free parameters which allow it to be fitted automatically to exponential functions. This is the first eighth order exponentially fitted method in the literature. Numerical results also indicate that the new method is much more accurate than other classical and exponentially fitted methods.

SOME LOW ORDER TWO-STEP ALMOST P-STABLE METHODS WITH PHASE-LAG OF ORDER INFINITY FOR THE NUMERICAL INTEGRATION OF THE RADIAL SCHRÖDINGER EQUATION

1995 ◽  
Vol 10 (16) ◽  
pp. 2431-2438 ◽  
Author(s):  
T.E. SIMOS
Keyword(s):  
Schrödinger Equation ◽  
Numerical Integration ◽  
Schrodinger Equation ◽  
Local Error ◽  
Phase Lag ◽  
New Methods ◽  
Step Procedure ◽  
Radial Schrödinger Equation ◽  
Algebraic Order ◽  
Local Error Estimate

Some two-step P-stable methods with phase-lag of order infinity are developed for the numerical integration of the radial Schrödinger equation. The methods are of O(h2) and O(h4) respectively. We produce, based on these methods and on a new local error estimate, a very simple variable step procedure. Extensive numerical testing indicates that these new methods are generally more accurate than other two-step methods with higher algebraic order.

SIMPLE AND ACCURATE EXPLICIT BESSEL AND NEUMANN FITTED METHODS FOR THE NUMERICAL SOLUTION OF THE SCHRÖDINGER EQUATION

2000 ◽  
Vol 11 (01) ◽  
pp. 79-89 ◽  
Author(s):  
T. E. SIMOS
Keyword(s):  
Schrödinger Equation ◽  
Numerical Solution ◽  
Schrodinger Equation ◽  
New Approach ◽  
New Methods ◽  
Radial Schrödinger Equation ◽  
Variable Step ◽  
Algebraic Order ◽  
Free Parameters ◽  
Step Algorithm

Explicit second and fourth algebraic order methods for the numerical solution of the Schrödinger equation are developed in this paper. The new methods have free parameters defined so that the methods are fitted to spherical Bessel and Neumann functions. Based on these new methods we obtained a variable-step algorithm. The results produced based on the numerical solution of the radial Schrödinger equation and the coupled differential equations arising from the Schrödinger equation indicate that this new approach is more efficient than other well known ones.

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