We construct a numerical scheme based on the scalar auxiliary variable (SAV) approach in time and the MAC discretization in space for the Cahn–Hilliard–Navier–Stokes phase- field model, prove its energy stability, and carry out error analysis for the corresponding Cahn–Hilliard–Stokes model only. The scheme is linear, second-order, unconditionally energy stable and can be implemented very efficiently. We establish second-order error estimates both in time and space for phase-field variable, chemical potential, velocity and pressure in different discrete norms for the Cahn–Hilliard–Stokes phase-field model. We also provide numerical experiments to verify our theoretical results and demonstrate the robustness and accuracy of our scheme.

中文翻译：

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关于 Cahn-Hilliard-Navier-Stokes 相场模型的 SAV-MAC 方案及其对应 Cahn-Hilliard-Stokes 情况的误差分析

针对Cahn-Hilliard-Navier-Stokes相场模型，我们构建了一个基于时间标量辅助变量（SAV）方法和空间MAC离散化的数值格式，证明了其能量稳定性，并对对应的误差进行了误差分析。仅限 Cahn-Hilliard-Stokes 模型。该方案是线性的、二阶的、无条件能量稳定的，并且可以非常有效地实现。我们在 Cahn-Hilliard-Stokes 相场模型的不同离散范数下建立了相场变量、化学势、速度和压力在时间和空间上的二阶误差估计。我们还提供数值实验来验证我们的理论结果并证明我们方案的鲁棒性和准确性。