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 second-order backward differentiation formulation (BDF2) scheme with variable time steps for gradient flow problems using a scalar auxiliary (SAV) approach. Comparing the traditional SAV approach (Shen et al. in J Comput Phys 353:407–416, 2018), only use first-order method to approximate variable. This treatment does not affect accuracy of unknown function $$\phi $$ , is essentially important deriving unconditional energy stability proposed BDF2 steps. We prove modified discrete adjacent step ratio $$\gamma _{n+1}:=\tau _{n+1}/\tau _{n}\le 4.8645$$ . The uniform $$H^{2}$$ bound numerical solution derived under mild regularity restriction on initial condition, that ({\varvec{x}},0)\in H^{2}$$ Based solution, rigorous error estimate carried out nonuniform temporal mesh. Finally, serval tests are provided validate theoretical claims. With application adaptive time-stepping strategy, efficiency our can be clearly observed coarsening dynamics simulation.