  Time-reversible symplectic methods, which are precisely compatible with
Liouville's phase-volume-conservation theorem, are often recommended for
computational simulations of Hamiltonian mechanics. Lack of energy drift is an
apparent advantage of such methods. But all numerical methods are susceptible
to Lyapunov instability, which severely limits the maximum time for which
chaotic solutions can be "accurate". The "advantages" of higher-order methods
are lost rapidly for typical chaotic Hamiltonians. We illustrate these
difficulties for a useful reproducible test case, the two-dimensional
one-particle cell model with specially smooth forces. This Hamiltonian problem
is chaotic and occurs on a three-dimensional constant-energy shell, the minimum
dimension for chaos. We benchmark the problem with quadruple-precision
trajectories using the fourth-order Candy-Rozmus, fifth-order Runge-Kutta, and
eighth-order Schlier-Seiter-Teloy integrators. We compare the last,
most-accurate particle trajectories to those from six double-precision
algorithms, four symplectic and two Runge-Kutta.
