The velocity errors of the classical marker and cell (MAC) scheme are dependent on the pressure approximation errors, which is non-pressure-robust and will cause the accuracy of the velocity approximation to deteriorate when the pressure approximation is poor. In this paper, we first propose the reconstructed MAC scheme (RMAC) based on the finite volume method to obtain the pressure-robustness for the time-dependent Stokes equations and then construct the $\mu$-robust and physics-preserving RMAC scheme on non-uniform grids for the Navier--Stokes equations, where $\mu$-robustness means that the velocity errors do not blow up for small viscosity $\mu$ when the true velocity is sufficiently smooth. Compared with the original MAC scheme, which was analyzed in [SIAM J. Numer. Anal. 55 (2017): 1135-1158], the RMAC scheme is different only on the right-hand side for Stokes equations. It can also be proved that the constructed scheme satisfies the local mass conservation law, the discrete unconditional energy dissipation law, the momentum conservation, and the angular momentum conservation for the Stokes and Navier--Stokes equations. Furthermore, by constructing the new auxiliary function depending on the velocity and using the high-order consistency analysis, we can obtain the pressure-robust and $\mu$-robust error estimates for the velocity and derive the second-order superconvergence for the velocity and pressure in the discrete $l^{\infty}(l^2)$ norm on non-uniform grids and the discrete $l^{\infty}(l^{\infty})$ norm on uniform grids. Finally, numerical experiments using the constructed schemes are demonstrated to show the robustness for our constructed schemes.

翻译：经典标记网格（MAC）格式的速度误差依赖于压力逼近误差，这种非压力鲁棒性会在压力逼近不佳时导致速度逼近精度下降。本文首先基于有限体积法提出重构MAC格式（RMAC），以获得瞬态Stokes方程的压力鲁棒性；随后针对Navier--Stokes方程在非均匀网格上构建具有$\mu$鲁棒性且物理保持的RMAC格式，其中$\mu$鲁棒性指当真实速度充分光滑时，速度误差不会因小黏度$\mu$而发散。相较于[SIAM J. Numer. Anal. 55 (2017): 1135-1158]中分析的原始MAC格式，RMAC格式仅对Stokes方程的右端项进行了修改。可证明所构建格式满足Stokes与Navier--Stokes方程的局部质量守恒律、离散无条件能量耗散律、动量守恒及角动量守恒。进一步地，通过构造依赖于速度的新辅助函数并采用高阶相容性分析，我们获得了速度的压力鲁棒与$\mu$鲁棒误差估计，并在非均匀网格的离散$l^{\infty}(l^2)$范数及均匀网格的离散$l^{\infty}(l^{\infty})$范数下，推导出速度与压力的二阶超收敛结果。最后，通过数值实验验证了所构建格式的鲁棒性。