  In this article we numerically revisit the long-time solution behavior of the
Camassa-Holm equation. The finite difference solution of this integrable
equation is sought subject to the newly derived initial condition with
Delta-function potential. Our underlying strategy of deriving a numerical phase
accurate finite difference scheme in time domain is to reduce the numerical
dispersion error through minimization of the derived discrepancy between the
numerical and exact modified wavenumbers. Additionally, to achieve the goal of
conserving Hamiltonians in the completely integrable equation of current
interest, a symplecticity-preserving time-stepping scheme is developed. Based
on the solutions computed from the temporally symplecticity-preserving and the
spatially wavenumber-preserving scheme, the long-time asymptotic CH solution
characters can be accurately depicted in distinct regions of the space-time
domain featuring with their own quantitatively very different solution
behaviors. We also aim to numerically confirm that in the two transition zones
their long-time asymptotics can indeed be described in terms of the
theoretically derived Painlev\'e transcendents. Another attempt of this study
is to numerically exhibit a close connection between the presently predicted
finite-difference solution and the solution of the Painlev\'e ordinary
differential equation of type II in two different transition zones.
