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A β-Factor for Stability: Provably Stable, High-Order Splitting Schemes for Incompressible Flows
Issue Vol. 1 No. 01 (2024): VOLUME01 ISSUE01 --- Section Articles
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Abstract
The numerical simulation of incompressible fluid dynamics, governed by the Navier-Stokes equations (NSEs), presents persistent challenges related to computational efficiency and numerical stability, especially for higher-order accurate methods. This paper introduces and analyzes a new class of fully decoupled, higher-order consistent splitting schemes for the incompressible Navier-Stokes equations. The core of our methodology is the formulation of schemes based on Taylor series expansions at a future time point tn+β, where β≥1 is a selectable free parameter. This approach generalizes the classical Backward Differentiation Formula (BDF) methods. The primary contribution of this work is a rigorous stability and error analysis for these schemes. We demonstrate that by selecting appropriate values for the parameter β—specifically, β=3 for the second-order scheme, β=6 for the third-order scheme, and β=9 for the fourth-order scheme—the resulting numerical solutions are uniformly bounded in a strong norm. This constitutes a proof of unconditional stability. Furthermore, we establish optimal global-in-time convergence rates for these schemes in both two and three-dimensional domains. To the best of our knowledge, these findings represent the first comprehensive stability and convergence results for any fully decoupled scheme for the Navier-Stokes equations with an order of accuracy higher than two. The theoretical analysis is substantiated by numerical experiments, which validate the unconditional stability of the new third- and fourth-order schemes. In contrast, we show that schemes based on the conventional BDF approach (i.e., β=1) are not unconditionally stable. The proposed schemes achieve their theoretically predicted orders of convergence, offering a robust and efficient pathway for high-fidelity simulations of incompressible flows..
