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 -semi-norm and velocity in the -norm are able to achieve second-order convergence rates in time and the in space. The nonlocal variables and also achieve the same convergence rate. In addition, the pressure in the -norm can only reach first-order convergence rate in time and the in space. At the same time, several numerical examples are given to illustrate the accuracy and effectiveness of the numerical scheme.

中文翻译：

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两相Cahn-Hilliard-Navier-Stokes不可压缩流Crank-Nicolson-Leapfrog格式的误差分析


本文严格证明了基于标量辅助变量（SAV）的两相Cahn-Hilliard-Navier-Stokes（CHNS）不可压缩流动方程的Crank-Nicolson-Leapfrog（CNLF）时间步进格式的误差估计。由于多变量的复杂性和方程的强耦合性，证明严格的误差估计并不容易。在相应的正则性假设和两个拟投影的负范数估计的超收敛下，我们证明了-半范数下的相场和-范数下的速度误差估计能够实现二阶收敛时间和空间上的速率。非局部变量 和 也达到相同的收敛速度。另外，-范数下的压力在时间和空间上只能达到一阶收敛速度。同时，给出了几个数值算例，说明了数值方案的准确性和有效性。