On the long-time stability of the implicit Euler scheme for the 2D space-periodic Navier–Stokes equations

Article type: Research Article

Authors: Tone, Florentina

Affiliations: Department of Mathematics and Statistics, University of West Florida, Pensacola, FL 32514 E-mail: [email protected]

Abstract: In this article we discretize the two-dimensional space-periodic Navier–Stokes equations in time using the implicit Euler scheme and, with the aid of the discrete Gronwall lemma and the uniform discrete lemma, we prove that the scheme is H2-uniformly stable in time. Moreover, in a final remark, we describe how the above result can be extended to show that the implicit Euler scheme is uniformly stable with respect to the H3-norm, for all time.

Keywords: Navier–Stokes equations, discrete Gronwall lemmas, implicit Euler scheme

Journal: Asymptotic Analysis, vol. 51, no. 3-4, pp. 231-245, 2007