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.

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.

High-algebraic, high-phase lag methods for accurate computations for the elastic- scattering phase shift problem

1998 ◽  
Vol 76 (6) ◽  
pp. 473-493 ◽  
Author(s):  
T E Simos
Keyword(s):  
Elastic Scattering ◽  
Phase Shift ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Finite Difference Methods ◽  
Phase Lag ◽  
New Methods ◽  
Shift Problem ◽  
Algebraic Order ◽  
Scattering Phase

A family of three new hybrid eighth-algebraic-order two-step methods with phase lag of order 16, 18, and 20 are developed for computing elastic-scattering phase shifts of the one-dimensional Schrödinger equation. Based on these new methods, we obtain some new embedded variable-step procedures for the numerical integration of the Schrödinger equation. Numerical results obtained for both the integration of the phase-shift problem for the well known case of the Lennard–Jones potential and the integration of coupled differential equation arising from the Schrödinger equation show that these new methods are better than other finite-difference methods. PACS Nos.: 02.00, 02.70, 03.00, 03.65

High Algebraic Order Methods with Minimal Phase-Lag for Accurate Solution of the Schrödinger Equation

1998 ◽  
Vol 09 (07) ◽  
pp. 1055-1071 ◽  
Author(s):  
T. E. Simos
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Finite Difference Methods ◽  
Accurate Solution ◽  
Phase Lag ◽  
Jones Potential ◽  
Coupled Equations ◽  
Radial Schrödinger Equation ◽  
Algebraic Order ◽  
Accurate Computations

A family of new hybrid four-step tenth algebraic order methods with phase-lag of order fourteen is developed for accurate computations of the radial Schrödinger equation. Numerical results obtained for the integration of the phase shift problem for the well known case of the Lennard-Jones potential and for the numerical solution of the coupled equations arising from the Schrödinger equation show that these new methods are better than other finite difference methods.

A NEW SYMMETRIC EIGHT-STEP PREDICTOR-CORRECTOR METHOD FOR THE NUMERICAL SOLUTION OF THE RADIAL SCHRÖDINGER EQUATION AND RELATED ORBITAL PROBLEMS

2011 ◽  
Vol 22 (02) ◽  
pp. 133-153 ◽  
Author(s):  
G. A. PANOPOULOS ◽  
Z. A. ANASTASSI ◽  
T. E. SIMOS
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
New Method ◽  
Phase Lag ◽  
Oscillating Solutions ◽  
Orbital Problems ◽  
Radial Schrödinger Equation ◽  
Algebraic Order ◽  
Predictor Corrector ◽  
Minimal Phase

A new general multistep predictor-corrector (PC) pair form is introduced for the numerical integration of second-order initial-value problems. Using this form, a new symmetric eight-step predictor-corrector method with minimal phase-lag and algebraic order ten is also constructed. The new method is based on the multistep symmetric method of Quinlan–Tremaine,1 with eight steps and 8th algebraic order and is constructed to solve numerically the radial time-independent Schrödinger equation. It can also be used to integrate related IVPs with oscillating solutions such as orbital problems. We compare the new method to some recently constructed optimized methods from the literature. We measure the efficiency of the methods and conclude that the new method with minimal phase-lag is the most efficient of all the compared methods and for all the problems solved.

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.

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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