Convergence analysis of a second-order SAV-ZEC scheme for the Cahn-Hilliard-Navier-Stokes system

Incorporating the scalar auxiliary variable (SAV) method and the zero energy contribution (ZEC) technique, we analyze a linear and fully decoupled numerical scheme for the Cahn-Hilliard-Naiver-Stokes (CHNS) system. More precisely, the fully discrete scheme combines the marker-and-cell (MAC) finite difference spatial approximation and BDF2 temporal discretization, as well as the Adams-Bashforth extrapolation for the nonlinear terms, based on the SAV-ZEC reformulation. A pressure correction approach is applied to decouple the Stokes equation. Only constant-coefficient Poisson-like solvers are needed in the implementation for the resulting numerical system. The numerical scheme is unconditionally stable with respect to a rewritten total energy functional, represented in terms of one auxiliary variable in the double-well potential, another auxiliary variable to balance all the nonlinear and coupled terms, the surface energy in the original phase variable, combined with the kinematic energy part. Specifically, the error estimate for the phase variable in the $\ell^{\infty}(0,T;H_h^1)\cap\ell^2(0,T;H_h^3)$ norm, the velocity variable in the $\ell^{\infty}(0,T;\ell^2)\cap\ell^2(0,T;H_h^1)$ norm, is derived with optimal convergence rates.