Error estimate of a consistent splitting GSAV scheme for the Navier-Stokes equations

TL;DR

This paper provides a rigorous error analysis of a linear, unconditionally stable semi-discrete scheme with a GSAV approach for Navier-Stokes equations, establishing optimal error estimates in 2D and 3D.

Contribution

It introduces a first-order semi-discrete splitting scheme with GSAV for Navier-Stokes, offering the first rigorous error estimates for this approach.

Findings

Optimal global error estimates in 2D

Optimal local error estimates in 3D

Scheme is linear, unconditionally stable, and efficient

Abstract

We carry out a rigorous error analysis of the first-order semi-discrete (in time) consistent splitting scheme coupled with a generalized scalar auxiliary variable (GSAV) approach for the Navier-Stokes equations with no-slip boundary conditions. The scheme is linear, unconditionally stable, and only requires solving a sequence of Poisson type equations at each time step. By using the build-in unconditional stability of the GSAV approach, we derive optimal global (resp. local) in time error estimates in the two (resp. three) dimensional case for the velocity and pressure approximations.