Long time stability of a classical efficient scheme for two-dimensional Navier-Stokes equations

Gottlieb, S; Tone, F; Wang, C; Wang, X; Wirosoetisno, D

Abstract

This paper considers the long-time stability property of a popular semi-implicit scheme for the two-dimensional incompressible Navier–Stokes equations in a periodic box that treats the viscous term implicitly and the nonlinear advection term explicitly. We consider both the semidiscrete (discrete in time but continuous in space) and fully discrete schemes with either Fourier Galerkin spectral or Fourier pseudospectral (collocation) methods. We prove that in all cases, the scheme is long time stable provided that the timestep is sufficiently small. The long time stability in the L2 and H1 norms further leads to the convergence of the global attractors and invariant measures of the scheme to those of the Navier–Stokes equations at vanishing timestep.

Citation

Gottlieb, S., Tone, F., Wang, C., Wang, X., & Wirosoetisno, D. (2012). Long time stability of a classical efficient scheme for two-dimensional Navier-Stokes equations. SIAM Journal on Numerical Analysis, 50(1), 126-150. https://doi.org/10.1137/110834901

• Journal Article Type: Article
• Publication Date: Jan 1, 2012
• Deposit Date: Feb 12, 2012
• Publicly Available Date: Mar 23, 2012
• Journal: SIAM Journal on Numerical Analysis
• Print ISSN: 0036-1429
• Electronic ISSN: 1095-7170
• Publisher: Society for Industrial and Applied Mathematics
• Peer Reviewed: Peer Reviewed
• Volume: 50
• Issue: 1
• Pages: 126-150
• DOI: https://doi.org/10.1137/110834901
• Keywords: Two-dimensional Navier–Stokes equations, Semi-implicit schemes, Global attractor,; Invariant measures, Spectral, Collocation.
• Public URL: https://durham-repository.worktribe.com/output/1506082