We construct several variational integrators—integrators based on a discrete variational principle—for systems with Lagrangians of the form L = L _ { A } + \epsilon L _ { B } , with \epsilon \ll 1 , where L _ { A } describes an integrable system .
These integrators exploit that \epsilon \ll 1 to increase their accuracy by constructing discrete Lagrangians based on the assumption that the integrator trajectory is close to that of the integrable system .
Several of the integrators we present are equivalent to well-known symplectic integrators for the equivalent perturbed Hamiltonian systems , but their construction and error analysis is significantly simpler in the variational framework .
One novel method we present , involving a weighted time-averaging of the perturbing terms , removes all errors from the integration at \mathcal { O } \left ( \epsilon \right ) .
This last method is implicit , and involves evaluating a potentially expensive time-integral , but for some systems and some error tolerances it can significantly outperform traditional simulation methods .

\PACS 95.10.Ce 45.10.Db 45.20.Jj 45.50.Pk