A variable time-step IMEX-BDF2 SAV scheme and its sharp error estimate for the Navier-Stokes equations

We generalize the implicit-explicit (IMEX) second-order backward difference (BDF2) scalar auxiliary variable (SAV) scheme for Navier-Stokes equation with periodic boundary conditions (Huang and Shen, SIAM J. Numer. Anal. 59 (2021) 2926-2954) to a variable time-step IMEX-BDF2 SAV scheme, and carry out a rigorous stability and convergence analysis. The key ingredients of our analysis are a new modified discrete Gronwall inequality, exploration of the discrete orthogonal convolution (DOC) kernels, and the unconditional stability of the proposed scheme. We derive global and local optimal H-1 error estimates in 2D and 3D, respectively. Our analysis provides a theoretical support for solving Navier-Stokes equations using variable time-step IMEX-BDF2 SAV schemes. We also design an adaptive time-stepping strategy, and provide ample numerical examples to confirm the effectiveness and efficiency of our proposed methods.