Closed Newton-Cotes Trigonometrically-Fitted Formulae for Numerical Integration of the Schrödinger Equation

2007 ◽  
Vol 3 (1) ◽  
pp. 45-57 ◽  
Author(s):  
T.E. Simos
Keyword(s):  
Numerical Integration ◽  
Symplectic Geometry ◽  
Efficient Solution ◽  
Symplectic Integrators ◽  
Multilayer Structures ◽  
Symplectic Methods ◽  
One Dimensional ◽  
One Step ◽  
Differential Methods ◽  
Symplectic Schemes

In this paper we investigate the connection between closed Newton-Cotes formulae, trigonometrically-fitted differential methods, symplectic integrators and efficient solution of the Schr¨odinger equation. Several one step symplectic integrators have been produced based on symplectic geometry, as one can see from the literature. However, the study of multistep symplectic integrators is very poor. Zhu et. al. [1] has studied the symplectic integrators and the well known open Newton-Cotes differential methods and as a result has presented the open Newton-Cotes differential methods as multilayer symplectic integrators. The construction of multistep symplectic integrators based on the open Newton-Cotes integration methods was investigated by Chiou and Wu [2]. In this paper we investigate the closed Newton-Cotes formulae and we write them as symplectic multilayer structures. We also develop trigonometrically-fitted symplectic methods which are based on the closed Newton-Cotes formulae. We apply the symplectic schemes to the well known one-dimensional Schr¨odinger equation in order to investigate the efficiency of the proposed method to these type of problems.

CLOSED NEWTON–COTES TRIGONOMETRICALLY-FITTED FORMULAE FOR LONG-TIME INTEGRATION

2003 ◽  
Vol 14 (08) ◽  
pp. 1061-1074 ◽  
Author(s):  
T. E. SIMOS
Keyword(s):  
Symplectic Geometry ◽  
Equations Of Motion ◽  
Time Integration ◽  
Symplectic Integrators ◽  
Symplectic Methods ◽  
Long Time ◽  
One Step ◽  
Long Time Integration ◽  
Differential Methods ◽  
Symplectic Schemes

The connection between closed Newton–Cotes, trigonometrically-fitted differential methods and symplectic integrators is investigated in this paper. It is known from the literature that several one-step symplectic integrators have been obtained based on symplectic geometry. However, the investigation of multistep symplectic integrators is very poor. Zhu et al.2 presented the well known open Newton–Cotes differential methods as multilayer symplectic integrators. Chiou and Wu2 also investigated the construction of multistep symplectic integrators based on the open Newton–Cotes integration methods. In this paper we investigate the closed Newton–Cotes formulae and we write them as symplectic multilayer structures. After this we construct trigonometrically-fitted symplectic methods which are based on the closed Newton–Cotes formulae. We apply the symplectic schemes in order to solve Hamilton's equations of motion which are linear in position and momentum. We observe that the Hamiltonian energy of the system remains almost constant as integration procceeds.

ACCURATELY CLOSED NEWTON–COTES TRIGONOMETRICALLY-FITTED FORMULAE FOR THE NUMERICAL SOLUTION OF THE SCHRÖDINGER EQUATION

2013 ◽  
Vol 24 (03) ◽  
pp. 1350014 ◽  
Author(s):  
T. E. SIMOS
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Symplectic Geometry ◽  
Differential Method ◽  
High Order ◽  
Symplectic Integrators ◽  
Resonance Problem ◽  
Radial Schrödinger Equation ◽  
One Step ◽  
Comparative Error

The investigation on the connection between: (1) closed Newton–Cotes formulae of high-order, (2) trigonometrically-fitted differential schemes and (3) symplectic integrators is presented in this paper. In the last decades, several one step symplectic methods were obtained based on symplectic geometry (see the appropriate literature). The investigation on multistep symplectic integrators is poor. In the present paper: (1) we study a trigonometrically-fitted high-order closed Newton–Cotes formula, (2) we investigate the necessary conditions in a general eight-step differential method to be presented as symplectic multilayer integrator, (3) we present a comparative error analysis in order to show the theoretical superiority of the present method, (4) we apply it to solve the resonance problem of the radial Schrödinger equation. Finally, remarks and conclusions on the efficiency of the new developed method are given which are based on the theoretical and numerical results.

High-Order Symplectic Schemes for Stochastic Hamiltonian Systems

2014 ◽  
Vol 16 (1) ◽  
pp. 169-200 ◽  
Author(s):  
Jian Deng ◽  
Cristina Anton ◽  
Yau Shu Wong
Keyword(s):  
Hamiltonian Systems ◽  
High Order ◽  
Symplectic Integrators ◽  
Symplectic Methods ◽  
Computational Tools ◽  
Fully Implicit ◽  
Stochastic Hamiltonian Systems ◽  
Hamiltonian Functions ◽  
Symplectic Schemes

AbstractThe construction of symplectic numerical schemes for stochastic Hamiltonian systems is studied. An approach based on generating functions method is proposed to generate the stochastic symplectic integration of any desired order. In general the proposed symplectic schemes are fully implicit, and they become computationally expensive for mean square orders greater than two. However, for stochastic Hamiltonian systems preserving Hamiltonian functions, the high-order symplectic methods have simpler forms than the explicit Taylor expansion schemes. A theoretical analysis of the convergence and numerical simulations are reported for several symplectic integrators. The numerical case studies confirm that the symplectic methods are efficient computational tools for long-term simulations.

A TWELFTH-ORDER FOUR-STEP FORMULA FOR THE NUMERICAL INTEGRATION OF THE ONE-DIMENSIONAL SCHRÖDINGER EQUATION

2003 ◽  
Vol 14 (08) ◽  
pp. 1087-1105 ◽  
Author(s):  
ZHONGCHENG WANG ◽  
YONGMING DAI
Keyword(s):  
Schrödinger Equation ◽  
Numerical Integration ◽  
Schrodinger Equation ◽  
Numerical Test ◽  
New Method ◽  
Step Method ◽  
One Dimensional ◽  
The Stability ◽  
The One ◽  
Order Four

A new twelfth-order four-step formula containing fourth derivatives for the numerical integration of the one-dimensional Schrödinger equation has been developed. It was found that by adding multi-derivative terms, the stability of a linear multi-step method can be improved and the interval of periodicity of this new method is larger than that of the Numerov's method. The numerical test shows that the new method is superior to the previous lower orders in both accuracy and efficiency and it is specially applied to the problem when an increasing accuracy is requested.

scholarly journals Higher-order accurate compact difference solutions for vibration problems of one dimensional continuous systems

2011 ◽  
pp. 771-794
Author(s):  
Mondher Yahiaoui
Keyword(s):  
Natural Frequencies ◽  
Continuous Systems ◽  
One Dimensional ◽  
First Order ◽  
Compact Difference ◽  
Transverse Vibrations ◽  
Seventh Order ◽  
One Step ◽  
Vibration Problems ◽  
Difference Methods

In this paper, we present a fourth-order accurate and a seventh-order accurate, one-step compact difference methods. These methods can be used to solve initial or boundaryvalue problems which can be modeled by a first-order linear system of differential equations. It is then shown in detail how these methods can be used to solve vibration problems of onedimensional continuous systems. Natural frequencies of a cantilever beam in transverse vibrations are computed and the results are compared to analytical ones to prove the high accuracy and efficiency of both methods. A comparison was also made to a finite element solution and the results have shown that both compact-difference methods yield more accurate values even with a reduced number of intervals.

QUASI SYMPLECTIC INTEGRATORS: AN ALTERNATIVE TO NOSÉ HOOVER DYNAMICS?

2005 ◽  
Vol 05 (02) ◽  
pp. L225-L232
Author(s):  
RICCARDO MANNELLA
Keyword(s):  
Molecular Dynamics ◽  
Molecular Dynamics Simulations ◽  
Numerical Integration ◽  
Canonical Ensemble ◽  
Symplectic Integrators ◽  
Langevin Equations ◽  
Dynamics Simulations

Quasi symplectic algorithms for the numerical integration of Langevin equations describing systems in a canonical ensemble are discussed. It is shown that they could be an alternative to molecular dynamics simulations done with a Nosé Hoover booster.

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