Forward and non-forward symplectic integrators in solving classical dynamics problems

Forward time step integrators are splitting algorithms with only positive splitting coefficients. When used in solving physical evolution equations, these positive coefficients correspond to positive time steps. Forward algorithms are essential for solving time-irreversible equations that cannot be evolved using backward time steps. However, forward integrators are also better in solving time-reversible equations of classical dynamics by tracking as closely as possible the physical trajectory. This work compares in detail various forward and non-forward fourth-order integrators using three, four, five and six force evaluations. In the case of solving the 2D Kepler orbit, all non-forward integrators are optimized by simply minimizing the size of their backward time steps.