تقرير
Long-term accuracy of numerical approximations of SPDEs with the stochastic Navier-Stokes equations as a paradigm
| العنوان: | Long-term accuracy of numerical approximations of SPDEs with the stochastic Navier-Stokes equations as a paradigm |
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| المؤلفون: | Glatt-Holtz, Nathan E., Mondaini, Cecilia F. |
| سنة النشر: | 2023 |
| المجموعة: | Computer Science Mathematics Mathematical Physics |
| مصطلحات موضوعية: | Mathematics - Numerical Analysis, Mathematical Physics, Mathematics - Analysis of PDEs, Mathematics - Probability, 60H15, 76M35, 65C30, 37L40, 37M25 |
| الوصف: | 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. Comment: 72 pages |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2302.01461 |
| رقم الأكسشن: | edsarx.2302.01461 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |