Reaching the equilibrium: Long-term stable approximations for stochastic non-Newtonian Stokes equations with transport noise

TL;DR

This paper introduces a new numerical algorithm for approximating stochastic non-Newtonian Stokes equations with transport noise, demonstrating long-term stability and analyzing invariant measures, with applications to power-law fluids.

Contribution

It develops a fully discrete, stable numerical scheme for stochastic non-Newtonian fluids and constructs invariant measures, advancing understanding of long-term behavior under transport noise.

Findings

Transport noise enhances energy dissipation.

Noise promotes mixing and vortex formation.

Invariant measures can be characterized explicitly.

Abstract

We propose and analyse a novel, fully discrete numerical algorithm for the approximation of the generalised Stokes system forced by transport noise -- a prototype model for non-Newtonian fluids including turbulence. Utilising the Gradient Discretisation Method, we show that the algorithm is long-term stable for a broad class of particular Gradient Discretisations. Building on the long-term stability and the derived continuity of the algorithm's solution operator, we construct two sequences of approximate invariant measures. At the moment, each sequence lacks one important feature: either the existence of a limit measure, or the invariance with respect to the discrete semigroup. We derive an abstract condition that merges both properties, recovering the existence of an invariant measure. We provide an example for which invariance and existence hold simultaneously, and characterise the invariant measure completely. We close the article by conducting two numerical experiments that show the influence of transport noise on the dynamics of power-law fluids; in particular, we find that transport noise enhances the dissipation of kinetic energy, the mixing of particles, as well as the size of vortices.