Trigonometrically Fitted Symplectic Integrators

4 Trigonometrically Fitted Symplectic Integrators. - Consider the Hamiltonian problems which are given by equation (3). 

We introduce the following n-stage modified Runge-Kutta-Nystrom method  

Chemical Modelling Applications and Theory, Volume 2 METHOD TFST 

In order for the above method to satisfy the symplecticness conditions (8) and the functions cos(vx), sin(vx) the following system of equations is obtained  

One of the family of solutions of the above system of equations is given by  

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In 34 the eigenvalue problem of the one-dimensional time-independent Schrodinger equation is studied. Exponentially fitted and trigonometrically fitted symplectic integrators are developed, by modification of the first and second order Yoshida symplectic methods. Numerical results are presented for the one-dimensional harmonic oscillator and Morse potential. 

Adapted, Exponentially Fitted and Trigonometrically Fitted Symplectic Integrators... 

Another powerful area is the construction of exponentialy fitted and trigonometrically fitted symplectic integrators (as we have seen in this review). Recently symplectic integrators have been applied for the numerical solution of the onedimensional and two-dimensional Schrodinger equations. 

Th. Monovasilis, Z. Kalogiratou and T. E. Simos, Families of third and fourth algebraic order trigonometrically fitted symplectic methods for the numerical integration of Hamiltonian systems. Computer Physics Communications, 2007, 177, 757 763. 

The authors in this paper present an explicit symplectic method for the numerical solution of the Schrodinger equation. A modified symplectic integrator with the trigonometrically fitted property which is based on this method is also produced. Our new methods are tested on the computation of the eigenvalues of the one-dimensional harmonic oscillator, the doubly anharmonic oscillator and the Morse potential. 

Th. Monovasilis and Z. Kalogiratou, Trigonometrically and Exponentially fitted Symplectic Methods of third order for the Numerical Integration of the Schrodinger Equation, Appl. Num. Anal. Comp. Math., 2005, 2(2), 238-244. 

In ref. 146 the authors study the numerical integration of Hamiltonian systems by symplectic and trigonometrically fitted (TF) symplectic methods. More specifically, the authors consider systems with separable Hamiltonian of the form ... 

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