Exponentially Fitted Symplectic Integrators

5 Exponentially Fitted Symplectic Integrators. - Consider the Hamiltonian 

We consider the n-stage modified Rimge-Kutta-Nystrom method given by equations (27)-(29). In order for the above method to satisfy the symplecticness conditions (8) and the fimction exp(vx) the following system of equations is obtained  

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

For small values of w the above formulae are subject to heavy cancellations. In this case we use the following Taylor series expansions  

---

J. M. Franco, Exponentially fitted symplectic integrators of RKN type for solving oscillatory problems. Computer Physics Communications, 2007, 177, 479-492. 

T. Mono Vasilis, Z. Kalogiratou and T. E. Simos, Trigonometrically fitted and exponentially fitted symplectic methods for the numerical integration of the Schrodinger equation,... 

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 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. 

H. Van de Vyver, A fourth-order symplectic exponentially fitted integrator, Comput. Phys. Comm., 2006, 174, 255-262. 

The combination of the property of exponential fitting and symplecticness in the construction of integrators for long interval integration. 

---