The primitive equations (PEs) of the atmosphere and the ocean without viscosity are considered. A 2.5D model is introduced, whose motivation is described in the Introduction. A set of nonlocal boundary conditions is proposed, and well-posedness is established for the flows linearized around a constant velocity stratified flow; homogeneous and nonhomogeneous boundary conditions are considered. A related model of dimension 2.5, of physical interest but with fewer degrees of freedom, is also considered at the end.
a b s t r a c tWe deal with the time-dependent Navier-Stokes equations (NSE) with Dirichlet boundary conditions on the whole domain or, on a part of the domain and open boundary conditions on the other part. It is shown numerically that combining the penalty-projection method with spatial discretization by the Marker And Cell scheme (MAC) yields reasonably good results for solving the above-mentioned problem. The scheme which has been introduced combines the backward difference formula of second-order (BDF2, namely Gear's scheme) for the temporal approximation, the second-order Richardson extrapolation for the nonlinear term, and the penalty-projection to split the velocity and pressure unknowns. Similarly to the results obtained for other projection methods, we estimate the errors for the velocity and pressure in adequate norms via the energy method. IntroductionSolving the time-dependent NSE for the incompressible fluid flow is still a major problem. The most important difficulty for solving such a problem in many physical applications is that the numerical simulation is CPU-time consuming. In fact, at each time step the velocity and pressure are coupled by the incompressibility constraint. There are various ways to discretize the time-dependent NSE; however, the most popular one is using projection methods, like pressure-correction methods. This family of methods has been introduced by Chorin and Temam [2,24] in the late sixties. These time-marching techniques are based on a splitting method that may be viewed as a predictor-corrector strategy aimed at uncoupling the viscous diffusion and incompressibility effects. Indeed, the interest in pressure-correction projection methods is due to the fact that the velocity and the pressure are computed separately. First, we solve the momentum balance equation to obtain an intermediate velocity and then, this predicted velocity is projected onto a space of solenoïdal vector fields. However, these prediction-correction methods introduce an additional numerical error, named the splitting error, which must be at worst of the same order as the time discretization error.A recent overview of several fractional techniques including pressure-correction and incremental projection methods can be found in [7].Moreover, Shen in [20] has introduced a modified approach which consists of constraining the divergence of the intermediate velocity field by adding in the first step of the scheme an augmentation term built from the divergence constraint, i.e. of the same form as in Augmented Lagrangian methods [4]. The divergence constraint is now treated both as 229 a penalty in the prediction step and by the projection step, which gives the name of the method. One can consider that the introduction of these penalty-projection schemes is an improvement of both pressure-correction and penalty methods (see [22] for an analysis of the penalty method).Recently some authors like Jobelin et al. proposed such a penalty-projection method by generalizing the prediction step with an augmentation para...
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