An implicit--explicit second order BDF numerical scheme with variable steps for gradient flows

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.