Robust superconvergence analysis of physics-preserving RMAC scheme for the Stokes and Navier--Stokes equations on non-uniform grids at high Reynolds numbers
Binghong Li, Xiaoli Li, Xu Li, Hongxing Rui
TL;DR
The paper tackles the vulnerability of velocity errors in the classic MAC scheme to pressure approximations by developing the RMAC scheme, a finite-volume reformulation that achieves pressure-robustness and $μ$-robustness for time-dependent Stokes and Navier–Stokes equations on non-uniform grids. By introducing velocity-dependent auxiliary functions and a gradient-invariance property, RMAC decomposes the problem into pressure-robust subproblems and ensures discrete conservation laws, including unconditional energy dissipation, momentum, and angular momentum, while maintaining high-order accuracy. The authors prove rigorous error estimates showing second-order superconvergence for velocity and pressure in non-uniform-grid $l^{∞}(l^2)$ norms and uniform-grid $l^{∞}(l^{∞})$ norms, supported by numerical experiments that validate robustness and convergence rates. The results offer a robust, physically faithful, and computationally efficient framework for simulating incompressible flows at high Reynolds numbers, with potential extensions to three dimensions.
Abstract
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 $μ$-robust and physics-preserving RMAC scheme on non-uniform grids for the Navier--Stokes equations, where $μ$-robustness means that the velocity errors do not blow up for small viscosity $μ$ 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 $μ$-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.