Accuracy and linear stability of RKN methods for solving second-order stiff problems

A general analysis of accuracy and linear stability of Runge–Kutta–Nyström (RKN) methods for solving second-order stiff problems is carried out. This analysis reveals that when components with large frequencies (stiff frequencies) and small amplitudes appear in the solution of the problem, the accuracy of an unconditionally stable RKN method can be seriously affected unless certain algebraic conditions are satisfied. Based on these algebraic conditions we derive new fourth-order A-stable diagonally implicit RKN (DIRKN) methods with different dispersion order and stage order. The numerical experiments carried out show the efficiency of the new methods when they are compared with other DIRKN codes proposed in the scientific literature for solving second-order stiff problems.