Sixth-order symmetric and symplectic exponentially fitted Runge–Kutta methods of the Gauss type

Abstract

The construction of exponentially fitted Runge–Kutta (EFRK) methods for the numerical integration of Hamiltonian systems with oscillatory solutions is considered. Based on the symplecticness, symmetry, and exponential fitting properties, two new three-stage RK integrators of the Gauss type with fixed or variable nodes, are obtained. The new exponentially fitted RK Gauss type methods 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 , and in particular { sin ( ω t ) , cos ( ω t ) } when λ = i ω , ω ∈ R . The algebraic order of the new integrators is also analyzed, obtaining that they are of sixth-order like the classical three-stage RK Gauss method. Some numerical experiments show that the new methods are more efficient than the symplectic RK Gauss methods (either standard or else exponentially fitted) proposed in the scientific literature.