Almost symplectic Runge–Kutta schemes for Hamiltonian systems

Symplectic Runge–Kutta schemes for the integration of general Hamiltonian systems are implicit. In practice, one has to solve the implicit algebraic equations using some iterative approximation method, in which case the resulting integration scheme is no longer symplectic. In this paper, the preservation of the symplectic structure is analyzed under two popular approximation schemes, fixed-point iteration and Newton’s method, respectively. Error bounds for the symplectic structure are established when N fixed-point iterations or N iterations of Newton’s method are used. The implications of these results for the implementation of symplectic methods are discussed and then explored through numerical examples. Numerical comparisons with non-symplectic Runge–Kutta methods and pseudo-symplectic methods are also presented.