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

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 stochastic partial differential equation 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.