Journal article
Long-term accuracy of numerical approximations of SPDEs with the stochastic Navier–Stokes equations as a paradigm
IMA journal of numerical analysis, v 45(3), pp 1648-1742
May 2025
Abstract
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 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.
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Details
- Title
- Long-term accuracy of numerical approximations of SPDEs with the stochastic Navier–Stokes equations as a paradigm
- Creators
- Nathan E Glatt-Holtz - Indiana University BloomingtonCecilia F Mondaini - Drexel University
- Publication Details
- IMA journal of numerical analysis, v 45(3), pp 1648-1742
- Publisher
- Oxford University Press
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:001271039100001
- Scopus ID
- 2-s2.0-105008173902
- Other Identifier
- 991021894215504721
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- Collaboration types
- Domestic collaboration
- Web of Science research areas
- Mathematics, Applied