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Long-Time H1-Stability of the Cauchy One-Leg Theta-Method for the Navier-Stokes Equations

Isabel Barrio Sanchez, Catalin Trenchea, Wenlong Pei

Abstract

In this paper we study the long-time stability of the Cauchy one-leg theta-methods for the two-dimensional NavierStokes equations. We establish the uniform dissipativity in H^1, in the sense that the semi-discrete-in-time approximations possess a global attractor for a small enough time step, using the discrete Gronwall lemma and the discrete uniform Gronwall lemma.

Long-Time H1-Stability of the Cauchy One-Leg Theta-Method for the Navier-Stokes Equations

Abstract

In this paper we study the long-time stability of the Cauchy one-leg theta-methods for the two-dimensional NavierStokes equations. We establish the uniform dissipativity in H^1, in the sense that the semi-discrete-in-time approximations possess a global attractor for a small enough time step, using the discrete Gronwall lemma and the discrete uniform Gronwall lemma.

Paper Structure

This paper contains 7 sections, 11 theorems, 299 equations.

Table of Contents

  1. Introduction and background
  2. Mathematical preliminaries
  3. H-Stability
  4. V-Stability
  5. Conclusions
  6. Funding
  7. Appendix: Computing $\alpha,\varepsilon, a,b$

Key Result

Lemma 1

For every $n\geq0$ and $\theta\in[0,1]$ we have

Theorems & Definitions (22)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Proposition 1
  • proof
  • Corollary 1
  • proof
  • Lemma 3
  • proof
  • ...and 12 more