Explicit exponentially fitted two-step hybrid methods of high order for second-order oscillatory IVPs

The construction of exponentially fitted (EF) two-step hybrid methods for the numerical integration of oscillatory second-order IVPs is analyzed. These methods integrate exactly differential systems whose solutions can be expressed as linear combinations of the set of functions {exp(λt),exp(−λt)},λ∈height6ptC, or equivalently {sin (ωt), cos (ωt  )} when λ=iω,ω∈IR, where λ represents an approximation of the main frequency of the problem. The necessary and sufficient conditions for this class of EF hybrid methods to have algebraic order p are derived. With the help of these order conditions and the EF conditions we construct explicit EF two-step hybrid methods with symmetric nodes and weights and orders six and seven. The numerical experiments carried out with several orbital and oscillatory problems show that the new high order EF two-step hybrid methods are more efficient than other EF and standard codes proposed in the scientific literature.