  We present an optimal rate convergence analysis for a second order accurate
in time, fully discrete finite difference scheme for the
Cahn-Hilliard-Navier-Stokes (CHNS) system, combined with logarithmic
Flory-Huggins energy potential. The numerical scheme has been recently
proposed, and the positivity-preserving property of the logarithmic arguments,
as well as the total energy stability, have been theoretically justified. In
this paper, we rigorously prove second order convergence of the proposed
numerical scheme, in both time and space. Since the CHNS is a coupled system,
the standard $\ell^\infty (0, T; \ell^2) \cap \ell^2 (0, T; H_h^2)$ error
estimate could not be easily derived, due to the lack of regularity to control
the numerical error associated with the coupled terms. Instead, the
$\ell^\infty (0, T; H_h^1) \cap \ell^2 (0, T; H_h^3)$ error analysis for the
phase variable and the $\ell^\infty (0, T; \ell^2)$ analysis for the velocity
vector, which shares the same regularity as the energy estimate, is more
suitable to pass through the nonlinear analysis for the error terms associated
with the coupled physical process. Furthermore, the highly nonlinear and
singular nature of the logarithmic error terms makes the convergence analysis
even more challenging, since a uniform distance between the numerical solution
and the singular limit values of is needed for the associated error estimate.
Many highly non-standard estimates, such as a higher order asymptotic expansion
of the numerical solution (up to the third order accuracy in time and fourth
order in space), combined with a rough error estimate (to establish the maximum
norm bound for the phase variable), as well as a refined error estimate, have
to be carried out to conclude the desired convergence result.
