  We investigate the difference between the velocity Verlet and the
Liouville-operator-derived (LOD) algorithms by studying two non-Hamiltonian
systems, one dissipative and the other conservative, for which the Jacobian of
the transformation can be determined exactly. For the two systems, we
demonstrate that (1) the velocity Verlet scheme fails to integrate the former
system while the first- and second-order LOD schemes succeed, (2) some
first-order LOD fails to integrate the latter system while the velocity Verlet
and the other first- and second-order schemes succeed. We have shown that the
LOD schemes are stable for the former system by determining the explicit forms
of the shadow Hamiltonians which are exactly conserved by the schemes. We have
shown that Jacobian of the velocity Verlet scheme for the former system and
that of the first-order LOD scheme for the latter system are always smaller
than the exact values, and therefore, the schemes are unstable. The
decomposition-order dependence of LOD schemes is also considered.
