Applications of extended simple equation method on unstable nonlinear Schrödinger equations

In the present paper we compare the two methodologies for the development of exponentially and trigonometrically fitted methods. One is based on the exact integration of the functions of the form: {1,x,x
 2,…,x
 p
 ,exp (±wx),xexp (±wx),…,x
 m
 exp (±w
 x)} and the second is based on the exact integration of the functions of the form: {1,x,x
 2,…,x
 p
 ,exp (±wx),exp (±2wx),…,exp (±mwx)}. The above functions are used in order to improve the efficiency of the classical methods of any kind (i.e. the method (5) with constant coefficients) for the numerical solution of ordinary differential equations of the form of the Schrodinger equation. We mention here that the above sets of exponential functions are the two most common sets of exponential functions for the development of the special methods for the efficient solution of the Schrodinger equation. It is first time in the literature in which the efficiency of the above sets of functions are studied and compared together for the approximate solution of the Schrodinger equation. We present the error analysis of the above two approaches for the numerical solution of the one-dimensional Schrodinger equation. Finally, numerical results for the resonance problem of the radial Schrodinger equation are presented.

Exponentially and Trigonometrically Fitted Methods for the Solution of the Schrödinger Equation

High order closed Newton–Cotes trigonometrically-fitted formulae for the numerical solution of the Schrödinger equation

In this paper, we investigate the connection between • closed Newton-Cotes formulae, • trigonometrically-fitted methods, • symplectic integrators and • efficient integration of the Schrodinger equation. The study of multistep symplectic integrators is very poor although in the last decades several one step symplectic integrators have been produced based on symplectic geometry (see the relevant literature and the references here). In this paper we study the closed Newton-Cotes formulae and we write them as symplectic multilayer structures. Based on the closed Newton-Cotes formulae, we also develop trigonometrically-fitted symplectic methods. An error analysis for the one-dimensional Schrodinger equation of the new developed methods and a comparison with previous developed methods is also given. We apply the new symplectic schemes to the well-known radial Schrodinger equation in order to investigate the efficiency of the proposed method to these type of problems.

High order closed Newton-Cotes trigonometrically-fitted formulae for the numerical solution of the Schrodinger equation

In this work we construct new Runge-Kutta-Nystrom methods especially designed to integrate exactly the test equation y^''=-w^2y. We modify two existing methods: the Runge-Kutta-Nystrom methods of fifth and sixth order. We apply the new methods to the computation of the eigenvalues of the Schrodinger equation with different potentials such as the harmonic oscillator, the doubly anharmonic oscillator and the exponential potential.

New modified Runge-Kutta-Nyström methods for the numerical integration of the Schrödinger equation

Symplectic integrators for the numerical solution of the Schrödinger equation

A new optimized two-step hybrid block method for the numerical integration of general second-order initial value problems is presented. The method considers two intra-step points which are selected adequately in order to optimize the local truncation errors of the main formulas for the solution and the derivative at the final point of the block. The new method is zero-stable and consistent with fifth algebraic order. Numerical experiments used revealed the superiority of the new method for solving this kind of problems, in comparison with methods of similar characteristics in the literature.

An optimized two-step hybrid block method for solving general second order initial-value problems

A family of trigonometrically fitted partitioned Runge–Kutta symplectic methods

In this work we consider symplectic Runge Kutta Nystr öm (SRKN) methods with three stages. We construct a fourth order SRKN with constant coefficients and a trigonometrically fitted SRKN method. We apply the new methods on the two-dimentional harmonic oscillator, the Stiefel-Bettis problem and on the computation of the eigenvalues of the Schr ödinger equation.

Th. Monovasilis

Papers

Symplectic integrators for the numerical solution of the Schrödinger equation

New modified Runge-Kutta-Nyström methods for the numerical integration of the Schrödinger equation

An optimized two-step hybrid block method for solving general second order initial-value problems

A family of trigonometrically fitted partitioned Runge–Kutta symplectic methods

Exponentially Fitted Symplectic Runge-Kutta-Nystr ¨om methods