  We focus on the numerical approximation of the Cahn-Hilliard type equations,
and present a family of second-order unconditionally energy-stable schemes. By
reformulating the equation into an equivalent system employing a scalar
auxiliary variable, we approximate the system at the time step $(n+\theta)$
($n$ denoting the time step index and $\theta$ is a real-valued parameter), and
devise a family of corresponding approximations that are second-order accurate
and unconditionally energy stable. This family of approximations contains the
often-used Crank-Nicolson scheme and the second-order backward differentiation
formula as particular cases. We further develop an efficient solution algorithm
for the resultant discrete system of equations to overcome the difficulty
caused by the unknown scalar auxiliary variable. The final algorithm requires
only the solution of four de-coupled individual Helmholtz type equations within
each time step, which involve only constant and time-independent coefficient
matrices that can be pre-computed. A number of numerical examples are presented
to demonstrate the performance of the family of schemes developed herein. We
note that this family of second-order approximations can be readily applied to
devise energy-stable schemes for other types of gradient flows when combined
with the auxiliary variable approaches.
