Time discrete approximation of weak solutions to stochastic equations of geophysical fluid dynamics and applications
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As a first step towards the numerical analysis of the stochastic primitive equations of the atmosphere and the oceans, the time discretization of these equations by an implicit Euler scheme is studied. From the deterministic point of view, the 3D primitive equations are studied in their full form on a general domain and with physically realistic boundary conditions. From the probabilistic viewpoint, this paper deals with a wide class of nonlinear, state dependent, white noise forcings which may be interpreted in either the Itô or the Stratonovich sense. The proof of convergence of the Euler scheme, which is carried out within an abstract framework, covers the equations for the oceans, the atmosphere, the coupled oceanic-atmospheric system as well as other related geophysical equations. The authors obtain the existence of solutions which are weak in both the PDE and probabilistic sense, a result which is new by itself to the best of our knowledge.
KeywordsNonlinear stochastic partial differential equations Geophysical fluid dynamics Primitive equations Discrete time approximation Martingale solutions Numerical analysis of stochastic PDEs
2000 MR Subject Classification35Q86 60H15 35Q35
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The authors have benefited from the hospitality of the department of mathematics Virginia Tech and from the Newton institute for mathematical sciences, University of Cambridge where the final stage of the writing was completed. The authors wish to thank Arnaud Debussche for his help with the use of the universally Radon measurable selection theorem. The authors also thank the referee(s) for bringing their attention to the article  which they regrettably overlooked.
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