  We construct high-order semi-discrete-in-time and fully discrete (with
Fourier-Galerkin in space) schemes for the incompressible Navier-Stokes
equations with periodic boundary conditions, and carry out corresponding error
analysis. The schemes are of implicit-explicit type based on a scalar auxiliary
variable (SAV) approach. It is shown that numerical solutions of these schemes
are uniformly bounded without any restriction on time step size. These uniform
bounds enable us to carry out a rigorous error analysis for the schemes up to
fifth-order in a unified form, and derive global error estimates in
$l^\infty(0,T;H^1)\cap l^2(0,T;H^2)$ in the two dimensional case as well as
local error estimates in $l^\infty(0,T;H^1)\cap l^2(0,T;H^2)$ in the three
dimensional case. We also present numerical results confirming our theoretical
convergence rates and demonstrating advantages of higher-order schemes for
flows with complex structures in the double shear layer problem.
