Symplectic Integrators with Complex Time Steps

Symplectic integrators are a popular tool applied to a variety of numerical problems. Many symplectic methods are known, but those higher than second order must include some substeps that travel backward compared with the main integration. To compensate for this, some substeps must have large coefficients. This produces large error terms and reduces the efficiency of high-order symplectic algorithms. The constraint equations for the substeps of high-order algorithms often have complex solutions in addition to the known real ones. The complex solutions typically generate symplectic integrators with small substeps, so these algorithms have substantially smaller error terms than conventional algorithms. This makes them more efficient in principle. Here third-, fourth-, and sixth-order algorithms are developed and tested for problems in which the Hamiltonian is split into kinetic and potential energy terms, and also for perturbed Kepler problems. In addition, symplectic correctors are developed for the third- and fourth-order methods. It is shown that complex integrators with leading error terms that have strictly imaginary coefficients effectively behave as if they are 1 order higher than expected.