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

Abstract

We generalize the implicit-explicit (IMEX) second-order backward difference (BDF2) scalar auxil- iary variable (SAV) scheme for Navier-Stokes equation with periodic boundary conditions [11, Huang and Shen, SIAM J. Numer. Anal., 2021] 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 Grönwall 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 solv- ing 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.

An efficient two-grid high-order compact difference scheme with variable-step BDF2 method for the semilinear parabolic equation

“…To analyze the variable-step BDF2 scheme for linear reactiondiffusion equations, Liao and Zhang [28] introduced a new concept, namely, discrete orthogonal convolution (DOC) kernels, and they improved the unconditional stability in the 𝐿 2 norm to 𝑟 𝑘 ≤ 3.561. Subsequently, with the help of DOC kernels and corresponding convolution inequalities, there is a great progress on the stability and error estimates of variable-step BDF2 method for nonlinear PDEs under 𝑟 𝑘 < 3.561 [31,33,42] and the further improved stepsize ratio restriction 𝑟 𝑘 < 4.8645 [11,12,26,32], respectively. Among all the variable-step BDF2 methods for nonlinear PDEs in the literature mentioned above, they treat the nonlinear terms fully or partially implicit, in which a nonlinear iteration must be implemented at each time step.…”