October 10 – Peter Minev: Recent splitting schemes for the incompressible Navier-Stokes equations

Peter Minev: Tuesday 10 October at 11 am, A415 Inria Paris.

The presentation will be focussed on two classes of recently developed splitting schemes for the Navier-Stokes equations.
The first class is based on the classical artificial compressibility (AC) method. The original method proposed by J. Shen in 1995 reduces the solution of the incompressible Navier-Stokes equations to a set of two or three parabolic problems in 2D and 3D correspondingly. Unfortunately, its accuracy is limited to first order in time and can be extended further only if the resulting scheme involves an elliptic problem for the velocity vector. Recently, together with J.L. Guermond (Texas A&M University) we proposed a scheme that extends the AC method to any order in time using a bootstrapping approach to the incompressibility constraint that essentially requires to solve only a set of parabolic equations for the velocity. The conditioning of the corresponding linear systems is therefore O(Δth^-2). This is generally better than solving a parabolic equation for the velocity and an elliptic equation for the pressure required by the various projection schemes that are perhaps the most popular approach at present. Besides, the bootstrapping algorithm allows to achieve any order in time, subject to some initialization conditions, in contrast to the projection methods whose accuracy seems to be essentially limited to second order on the velocity in the L2 norm.
The second class of methods is based on a novel approach to the Navier-Stokes equations that reformulates them in terms of stress variables. It was developed in a recent paper together with P. Vabishchevich (Russian Academy of Sciences). The main advantage of such an approach becomes clear when it is applied to fluid-structure interaction problems since in such case the problems for the fluid and the structure, both written in terms of stress variables, become very similar. Although at first glance the resulting tensorial problem is more difficult, if it is combined with a proper splitting, it yields locally one dimensional schemes with attractive properties, that are very competitive to the the most widely used schemes for the formulation in primitive variables. Several schemes for discretization of this formulation will be presented together with their stability analysis.
Finally, numerical results for a problem with a manufactured solution will be presented.

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