  We present a new numerical algorithm for evolving the Mixmaster spacetimes.
By using symplectic integration techniques to take advantage of the exact Taub
solution for the scattering between asymptotic Kasner regimes, we evolve these
spacetimes with higher accuracy using much larger time steps than previously
possible. The longer Mixmaster evolution thus allowed enables detailed
comparison with the Belinskii, Khalatnikov, Lifshitz (BKL) approximate
Mixmaster dynamics. In particular, we show that errors between the BKL
prediction and the measured parameters early in the simulation can be
eliminated by relaxing the BKL assumptions to yield an improved map. The
improved map has different predictions for vacuum Bianchi Type IX and magnetic
Bianchi Type VI$_0$ Mixmaster models which are clearly matched in the
simulation.
