A family of explicit parallel Runge–Kutta–Nyström methods

A procedure for the construction of high-order explicit parallel Runge–Kutta–Nyström (RKN) methods for solving second-order nonstiff initial value problems (IVPs) is analyzed. The analysis reveals that starting the procedure with a reference symmetric RKN method it is possible to construct high-order RKN schemes which can be implemented in parallel on a small number of processors. These schemes are defined by means of a convex combination of k disjoint si-stage explicit RKN methods which are constructed by connecting si steps of a reference explicit symmetric method. Based on the reference second-order Störmer–Verlet methods we derive a family of high-order explicit parallel schemes which can be implemented in variable-step codes without additional cost. The numerical experiments carried out show that the new parallel schemes are more efficient than some sequential and parallel codes proposed in the scientific literature for solving second-order nonstiff IVPs.