  In this paper, we propose and analyze an efficient implicit--explicit (IMEX)
second order in time backward differentiation formulation (BDF2) scheme with
variable time steps for gradient flow problems using the scalar auxiliary
variable (SAV) approach. We prove the unconditional energy stability of the
scheme for a modified discrete energy with the adjacent time step ratio
$\gamma_{n+1}:=\Dt_{n+1}/\Dt_{n}\leq 4.8645$. The uniform $H^{2}$ bound for the
numerical solution is derived under a mild regularity restriction on the
initial condition, that is $\phi(\x,0)\in H^{2}$. Based on this uniform bound,
a rigorous error estimate of the numerical solution is carried out on the
temporal nonuniform mesh. Finally, serval numerical tests are provided to
validate the theoretical claims. With the application of an adaptive
time-stepping strategy, the efficiency of our proposed scheme can be clearly
observed in the coarsening dynamics simulation.
