On correctors of symplectic integrators

An intensive discussion is given here with numerical illustration on the symplectic correctors proposed by Wisdom et al. (1996). A simple method is given for deriving the first and second order correctors of any symplectic integrators in terms of Lie series for the general case, in which the Hamiltonian can be separated into a main integrable part and several smaller integrable parts. The method is numerically applied to the 3-body system Sun-Jupiter-Saturn with the Hamiltonian of the form introduced by Duncan et al. (1998). It is shown that the first-order corrector is more precise that the uncorrected symplectic integrator. As to precision, numerical stability and computer efficiency, the first-order corrector is greatly superior to the second-order corrector when a large stepsize is adopted.