Stability and error analysis of a class of high-order IMEX schemes for Navier-stokes equations with periodic boundary conditions

Abstract: 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.