  In this paper we propose and analyze an energy stable numerical scheme for
the Cahn-Hilliard equation, with second order accuracy in time and the fourth
order finite difference approximation in space. In particular, the truncation
error for the long stencil fourth order finite difference approximation, over a
uniform numerical grid with a periodic boundary condition, is analyzed, via the
help of discrete Fourier analysis instead of the the standard Taylor expansion.
This in turn results in a reduced regularity requirement for the test function.
In the temporal approximation, we apply a second order BDF stencil, combined
with a second order extrapolation formula applied to the concave diffusion
term, as well as a second order artificial Douglas-Dupont regularization term,
for the sake of energy stability. As a result, the unique solvability, energy
stability are established for the proposed numerical scheme, and an optimal
rate convergence analysis is derived in the $\ell^\infty (0,T; \ell^2) \cap
\ell^2 (0,T; H_h^2)$ norm. A few numerical experiments are presented, which
confirm the robustness and accuracy of the proposed scheme.
