Exponentially Fitted and Trigonometrically Fitted Explicit Modified Runge-Kutta Type Methods for Solving y′′′x=fx,y,y′

Exponentially Fitted Symplectic Runge-Kutta-Nyström methods

In this work we consider symplectic Runge Kutta Nyström (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 Schrödinger equation.

Fourth-order symplectic exponentially-fitted modified Runge-Kutta methods of the Gauss type: a review

The construction of symmetric and symplectic exponentially-fitted Runge-Kutta methods for the numerical integration of Hamiltonian systems with oscillatory solutions is reconsidered. In previous papers fourth-order and sixth-order symplectic exponentially-fitted integrators of Gauss type, either with fixed or variable nodes, have been derived. In this paper new fourth-order integrators are cons...

A New Family of Phase-Fitted and Amplification-Fitted Runge-Kutta Type Methods for Oscillators

In order to solve initial value problems of differential equations with oscillatory solutions, this paper improves traditional Runge-Kutta RK methods by introducing frequency-depending weights in the update. New practical RK integrators are obtained with the phase-fitting and amplification-fitting conditions and algebraic order conditions. Two of the new methods have updates that are also phase...

Three-stage two-parameter symplectic, symmetric exponentially-fitted Runge-Kutta methods of Gauss type

We construct an exponentially-fitted variant of the well-known three stage Runge-Kutta method of Gauss-type. The new method is symmetric and symplectic by construction and it contains two parameters, which can be tuned to the problem at hand. Some numerical experiments are given.

An Embedded 4(3) Pair of Explicit Trigonometrically-Fitted Runge-Kutta-Nyström Method for Solving Periodic Initial Value Problems