Trigonometrically fitted multi-step RKN methods for second-order oscillatory initial value problems

Trigonometrically fitted multi-step Runge-Kutta-Nyström (TFMSRKN) methods for solving numerically oscillatory special second-order initial value problems are introduced. TFMSRKN methods integrate exactly the differential system whose solutions can be expressed as the linear combinations of functions from the set {exp(iwt),exp(−iwt)} or equivalently the set {cos (wt), sin (wt)}, where w represents an approximation of the main frequency of the problem. The corresponding order conditions are given and two explicit four-stage TFMSRKN methods with order five are constructed. Stability of the new methods is examined and the regions of stability are depicted. Numerical results show that our new methods are more efficient in comparison with other well-known high quality methods proposed in the scientific literature.