Long-term accuracy of numerical approximations of SPDEs with the stochastic Navier-Stokes equations as a paradigm

TL;DR

This paper develops a framework for analyzing the long-term accuracy of numerical approximations of SPDEs, exemplified by the stochastic 2D Navier-Stokes equations, focusing on Wasserstein contraction and invariant measure approximation.

Contribution

It introduces a new approach using refined weak Harris theorems to establish long-time accuracy bounds for SPDE approximations, applicable to systems with weaker dissipation or stronger nonlinearity.

Findings

Quantitative estimates on invariant measure approximation

Weak consistency bounds on approximation errors over time

Refined $L^2_x$ accuracy bounds for numerical schemes

Abstract

This work introduces a general framework for establishing the long time accuracy for approximations of Markovian dynamical systems on separable Banach spaces. Our results illuminate the role that a certain uniformity in Wasserstein contraction rates for the approximating dynamics bears on long time accuracy estimates. In particular, our approach yields weak consistency bounds on $\mathbb{R}^+$ while providing a means to sidestepping a commonly occurring situation where certain higher order moment bounds are unavailable for the approximating dynamics. Additionally, to facilitate the analytical core of our approach, we develop a refinement of certain `weak Harris theorems'. This extension expands the scope of applicability of such Wasserstein contraction estimates to a variety of interesting SPDE examples involving weaker dissipation or stronger nonlinearity than would be covered by the existing literature. As a guiding and paradigmatic example, we apply our formalism to the stochastic 2D Navier-Stokes equations and to a semi-implicit in time and spectral Galerkin in space numerical approximation of this system. In the case of a numerical approximation, we establish quantitative estimates on the approximation of invariant measures as well as prove weak consistency on $\mathbb{R}^+$. To develop these numerical analysis results, we provide a refinement of $L^2_x$ accuracy bounds in comparison to the existing literature which are results of independent interest.