An Implicit–Explicit Second-Order BDF Numerical Scheme with Variable Steps for Gradient Flows

Abstract

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 variable (SAV) approach. Comparing with the traditional second-order SAV approach (Shen et al. in J Comput Phys 353:407–416, 2018), we only use a first-order method to approximate the auxiliary variable. This treatment does not affect the second-order accuracy of the unknown function $$\phi $$ , and is essentially important for deriving the unconditional energy stability of the proposed BDF2 scheme with variable time steps. We prove the unconditional energy stability of the scheme for a modified discrete energy with the adjacent time step ratio $$\gamma _{n+1}:=\tau _{n+1}/\tau _{n}\le 4.8645$$ . The uniform $$H^{2}$$ bound for the numerical solution is derived under a mild regularity restriction on the initial condition, that is $$\phi ({\varvec{x}},0)\in H^{2}$$ . Based on this uniform bound of the numerical solution, a rigorous error estimate of the proposed scheme is carried out on the nonuniform temporal 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.