New methods for oscillatory systems based on ARKN methods

A one-parameter family of explicit fourth-order methods for oscillatory systems of the form y″+Ky=f(t,y), K being a symmetric positive semi-definite matrix, is obtained. The new methods possess eighth order for the unperturbed problem (f(t,y)≡0), and the free parameter is chosen so that the dispersion or the dissipation are optimized. The stability and phase properties of the new methods are analyzed, obtaining generalized stability regions for the second-order homogeneous linear test model. The numerical experiments carried out show that the new methods are very competitive when they are compared with standard, symplectic and special codes proposed in the scientific literature.