| Abstract: | In this paper we propose some efficient schemes for the Navier-Stokes equations. The proposed schemes are constructed based on an auxiliary variable reformulation of the underlying equations, recently introduced by Li et al. [20]. Our objectiveis to construct and analyze improved schemes, which overcome some of the shortcomings of the existing schemes. In particular, our new schemes have the capability to capture steady solutions for large Reynolds numbers and time step sizes, while keepingthe error analysis available. The novelty of our method is twofold: i) Use the Uzawaalgorithm to decouple the pressure and the velocity. This is to replace the pressure-correction method considered in [20]. ii) Inspired by the paper [21], we modify thealgorithm using an ingredient to capture stationary solutions. In all cases we analyze a first- and second-order schemes and prove the unconditionally energy stability.We also provide an error analysis for the first-order scheme. Finally we validate ourschemes by performing simulations of the Kovasznay flow and double lid driven cavity flow. These flow simulations at high Reynolds numbers demonstrate the robustnessand efficiency of the proposed schemes. |
