  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, fourth,
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.
