The authors examine a connection hetween symplectic methods and conventional numerical integration schemes, and use the Euler-Maclaurin summation formula to achieve leading-order correctors for those schemes. New symplectic schemes are synthesized, and their correctors are exploited to derive higher-order schemes. The suppression of their leading second-order error in the perturbing Hamiltonian is accomplished by using a modified potential. The authors also present a general formalism for the description of a class of symplectic schemes, and demonstrate the accuracy of two schemes with numerical results.