Stability and error analysis of a new class of higher-order consistent splitting schemes for the Navier-Stokes equations

TL;DR

This paper introduces a new class of higher-order, fully decoupled splitting schemes for the Navier-Stokes equations, demonstrating their stability and optimal convergence in 2D and 3D, with numerical validation.

Contribution

It presents the first stability and convergence analysis for fully decoupled, higher-than-second-order schemes for Navier-Stokes equations using a novel Taylor expansion approach.

Findings

Higher-order schemes with specific parameters are unconditionally stable.

Numerical results confirm optimal convergence rates in 2D and 3D.

Standard BDF-based schemes are not unconditionally stable, unlike the new schemes.

Abstract

A new class of fully decoupled consistent splitting schemes for the Navier-Stokes equations are constructed and analyzed in this paper. The schemes are based on the Taylor expansion at $t^{n+\beta}$ with $\beta\ge 1$ being a free parameter. It is shown that by choosing {\color{black} $\beta= 3, \,6,\,9$} respectively for the second-, third- and fourth-order schemes, their numerical solutions are uniformed bounded in a strong norm, and admit optimal global-in-time convergence rates in both 2D and 3D. {\color{black}These } results are the first stability and convergence results for any fully decoupled, higher than second-order schemes for the Navier-Stokes equations. Numerical results are provided to show that the third- and fourth-order schemes based on the usual BDF (i.e. $\beta=1$) are not unconditionally stable while the new third- and fourth-order schemes with suitable $\beta$ are unconditionally stable and lead to expected convergence rates.