A SYMMETRIC HIGH ORDER METHOD WITH MINIMAL PHASE-LAG FOR THE NUMERICAL SOLUTION OF THE SCHRÖDINGER EQUATION

2001 ◽  
Vol 12 (07) ◽  
pp. 1035-1042 ◽  
Author(s):  
T. E. SIMOS ◽  
JESUS VIGO AGUIAR
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
New Method ◽  
Phase Lag ◽  
Step Method ◽  
High Order Method ◽  
Radial Schrödinger Equation ◽  
Algebraic Order ◽  
Minimal Phase ◽  
Symmetric Method

In this paper, a new high algebraic order symmetric eight-step method is introduced. For this method, a direct formula for the computation of the phase-lag is given. Based on this formula, an eight-step symmetric method with minimal phase-lag is developed. The new method has better stability properties than the classical one. Numerical illustrations on the radial Schrödinger equation indicate that the new method is more efficient than older ones.

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.

Hybrid high algebraic order two-step method with vanished phase-lag and its first, second, third, fourth and fifth derivatives

2016 ◽  
Vol 27 (05) ◽  
pp. 1650049 ◽  
Author(s):  
Junyan Ma ◽  
T. E. Simos
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Truncation Error ◽  
Resonance Problem ◽  
Phase Lag ◽  
Step Method ◽  
Algebraic Order ◽  
Test Equation ◽  
The Stability ◽  
Scalar Test Equation

A hybrid tenth algebraic order two-step method with vanished phase-lag and its first, second, third, fourth and fifth derivatives are obtained in this paper. We will investigate • the construction of the method • the local truncation error (LTE) of the newly obtained method. We will also compare the lte of the newly developed method with other methods in the literature (this is called the comparative LTE analysis) • the stability (interval of periodicity) of the produced method using frequency for the scalar test equation different from the frequency used in the scalar test equation for phase-lag analysis (this is called stability analysis) • the application of the newly obtained method to the resonance problem of the Schrödinger equation. We will compare its effectiveness with the efficiency of other known methods in the literature. It will be proved that the developed method is effective for the approximate solution of the Schrödinger equation and related periodical or oscillatory initial value or boundary value problems.

A NEW NUMEROV-TYPE METHOD FOR COMPUTING EIGENVALUES AND RESONANCES OF THE RADIAL SCHRÖDINGER EQUATION

1996 ◽  
Vol 07 (01) ◽  
pp. 33-41 ◽  
Author(s):  
T. E. SIMOS
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Numerical Results ◽  
New Method ◽  
Resonance Problem ◽  
Step Method ◽  
Type Method ◽  
Radial Schrödinger Equation ◽  
Better Than

A two-step method is developed for computing eigenvalues and resonances of the radial Schrödinger equation. Numerical results obtained for the integration of the eigenvalue and the resonance problem for several potentials show that this new method is better than other similar methods.

A NEW METHODOLOGY FOR THE CONSTRUCTION OF OPTIMIZED RUNGE–KUTTA–NYSTRÖM METHODS

2011 ◽  
Vol 22 (06) ◽  
pp. 623-634 ◽  
Author(s):  
D. F. PAPADOPOULOS ◽  
T. E. SIMOS
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
First Integrals ◽  
New Method ◽  
Phase Lag ◽  
Nyström Methods ◽  
Algebraic Order ◽  
Runge Kutta ◽  
First Time ◽  
Zero Phase

In this paper, a new Runge–Kutta–Nyström method of fourth algebraic order is developed. The new method has zero phase-lag, zero amplification error and zero first integrals of the previous properties. Numerical results indicate that the new method is very efficient for solving numerically the Schrödinger equation. We note that for the first time in the literature we use the requirement of vanishing the first integrals of phase-lag and amplification error in the construction of efficient methods for the numerical solution of the Schrödinger equation.

scholarly journals An Efficient Numerical Method for the Solution of the Schrödinger Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-20 ◽  
Author(s):  
Licheng Zhang ◽  
Theodore E. Simos
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Truncation Error ◽  
New Method ◽  
Phase Lag ◽  
Local Truncation Error ◽  
Periodicity Analysis ◽  
Algebraic Order ◽  
Test Equation ◽  
Scalar Test Equation

The development of a new five-stage symmetric two-step fourteenth-algebraic order method with vanished phase-lag and its first, second, and third derivatives is presented in this paper for the first time in the literature. More specifically we will study(1)the development of the new method,(2)the determination of the local truncation error (LTE) of the new method,(3)the local truncation error analysis which will be based on test equation which is the radial time independent Schrödinger equation,(4)the stability and the interval of periodicity analysis of the new developed method which will be based on a scalar test equation with frequency different than the frequency of the scalar test equation used for the phase-lag analysis, and(5)the efficiency of the new obtained method based on its application to the coupled Schrödinger equations.

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.

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