Symmetric and symplectic exponentially fitted Runge–Kutta methods of high order

The construction of high order symmetric, symplectic and exponentially fitted Runge–Kutta (RK) methods for the numerical integration of Hamiltonian systems with oscillatory solutions is analyzed. Based on the symplecticness, symmetry, and exponential fitting properties, three new four-stage RK integrators, either with fixed- or variable-nodes, are constructed. The algebraic order of the new integrators is also studied, showing that they possess eighth-order of accuracy as the classical four-stage RK Gauss method. Numerical experiments with some oscillatory test problems are presented to show that the new methods are more efficient than other symplectic four-stage eighth-order RK Gauss codes proposed in the scientific literature.