Error analysis of Crank-Nicolson-Leapfrog scheme for the two-phase Cahn-Hilliard-Navier-Stokes incompressible flows

In this paper, the error estimates of the Crank-Nicolson-Leapfrog (CNLF) time-stepping scheme for the two-phase Cahn-Hilliard-Navier-Stokes (CHNS) incompressible flow equations based on scalar auxiliary variable (SAV) are strictly proved. Due to the complexity of the multiple variables and the strong coupling of the equations, it is not easy to prove rigorous error estimates. Under the corresponding regularity assumption and the superconvergence of the negative norm estimates of the two quasi-projections, we prove that the error estimates for phase-field in the 1-semi-norm and velocity u in the 1-norm are able to achieve second-order convergence rates in time and the ( +1 ) ( >= 1) in space. The nonlocal variables and also achieve the same convergence rate. In addition, the pressure in the 2-norm can only reach first-order convergence rate in time and the () ( ) ( >= 1) in space. At the same time, several numerical examples are given to illustrate the accuracy and effectiveness of the numerical scheme.