Exponentially fitted symplectic integrators of RKN type for solving oscillatory problems

Abstract

A class of explicit modified Runge–Kutta–Nyström (RKN) methods for the numerical integration of second-order IVPs with oscillatory solutions is presented. The symplecticness conditions and the exponential fitting conditions for this class of methods are derived. Based on this conditions, explicit modified RKN integrators with two and three stages per step which have algebraic orders two and four, respectively, are constructed. These new integrators preserve symplecticness when they are applied to Hamiltonian problems, and they integrate exactly differential systems whose solutions can be expressed as linear combinations of the set of functions { exp ( λ t ) , exp ( − λ t ) } , λ ∈ C , or equivalently { sin ( ω t ) , cos ( ω t ) } when λ = i ω , ω ∈ R . We also analyze the stability properties of the new integrators, obtaining generalized periodicity regions for the classical second-order linear test model. The numerical experiments carried out show that the new methods are more efficient than other symplectic and exponentially fitted codes proposed in the scientific literature.