New P-stable high-order methods with minimal phase-lag for the numerical integration of the radial Schrödinger equation

1997 ◽  
Vol 55 (6) ◽  
pp. 644-650 ◽  
Author(s):  
T E Simos
Keyword(s):  
Schrödinger Equation ◽  
Numerical Integration ◽  
Schrodinger Equation ◽  
High Order ◽  
Phase Lag ◽  
High Order Methods ◽  
Radial Schrödinger Equation ◽  
Minimal Phase

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.

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.

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