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

AbstractIn this work we construct new Runge–Kutta–Nyström methods especially designed to integrate exactly the test equation y″=−w2y. We modify two existing methods: the Runge–Kutta–Nyström methods of fifth and sixth order. We apply the new methods to the computation of the eigenvalues of the Schrödinger equation with different potentials such as the harmonic oscillator, the doubly anharmonic oscillator and the exponential potential

Exponentially fitted symplectic Runge-Kutta-Nyström methods

In this work we consider symplectic Runge Kutta Nystr¨om (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¨odinger equation

Symplectic Runge-Kutta-Nystr?m Methods with Phase-Lag Oder 8 and Infinity

In this work we consider Symplectic Runge Kutta Nystr¨om methods with five stages. A new fourth algebraic order method with phase-lag order eight is presented. Also the symplectic Runge Kutta Nystr¨om of Calvo and Sanz Serna with five stages and fourth order is modified to produce a phase-fitted method. We apply the new methods on several Hamiltonian systems and on the computation of the eigenvalues of the Schr¨odinger Equation

Construction of exponentially fitted symplectic Runge-Kutta-Nyström methods from partitioned Runge-Kutta methods

In this work we derive exponentially fitted symplectic Runge-Kutta-Nystr¨om (RKN) methods from symplectic exponentially fitted partitioned Runge-Kutta (PRK) methods methods. We construct RKN methods from PRK methods with up to five stages and fourth algebraic order