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

TL;DR

This paper analyzes the long-term stability of a numerical method for the 2D Navier-Stokes equations, proving uniform dissipativity and existence of a global attractor for small time steps.

Contribution

It establishes the H^1-stability and long-time behavior of the Cauchy one-leg theta-method for the Navier-Stokes equations, a novel stability result.

Findings

Proves uniform dissipativity in H^1 for the method.

Shows the existence of a global attractor for small time steps.

Uses discrete Gronwall lemmas to establish stability.

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.