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Section: New Results

High order approximation of the two fluid Braginskii model

Participants : Sebastian Minjeaud, Richard Pasquetti.

We work on a two fluid physical model developed in close connection with Ph. Ghendrih (IRFM). It is based on the electrostatic assumption, i.e. the magnetic field is given (the magnetic field induced by the plasma itself is negligible), on the hypothesis of electroneutrality (the density of ions and electrons are proportional) and on the Braginskii closure. On the basis of the conservation equations of density, electron and ion velocities, electron and ion temperatures and electrical charges, a set of 10 non-linear and strongly anisotropic coupled partial differential equations (PDE) can be set up. A high order Fourier-SEM (Spectral Element Method) code is currently developed to address this problem. This Fourier-SEM code is close to be operational for the full set of PDEs in a 3D toroidal geometry. The torus section is discretized with quadrangular elements and Fourier expansions are used in the toroidal direction. In time one uses an RK3 (third order Runge-Kutta) IMEX (Implicit-Explicit), so that the Lorentz terms are handled implicitly. The capability of this code to handle a strongly anisotropic diffusion in a 3D toroidal geometry has already been tested. The Braginskii closure has been implemented. The Bohm boundary conditions at the plates are also considered. In 2013, we worked on a parallel version of the code and on the robustness of our algorithms, to be able to make long time computations, e.g. a few hundreds of thermal times.

Parallelization of the full Braginskii (FBGKI) code.

A first parallel version of the FBGKI code is now operational. Tests were made on the Computional center of the University of Nice-Sophia Antipolis. Tests on a large number of processors have however not yet been carried out, since presently our goal is to improve the robustness of our algorithms. Our parallelization strategy is based on a domain decomposition technique in the poloidal plane, where the spectral element approximation is local. On the contrary, in toroidal direction the approximation is global since based on Fourier expansions.

Numerical stabilization.

A spectral vanishing viscosity (SVV) technique was implemented in the sequential version of the code. Such a stabilization technique relies on the idea of introducing viscosity in the high frequency range of the spectral element / Fourier approximation. Such an approach was first proposed for hyperbolic problems, typically the Burgers equation (E. Tadmor, 1989). Later on, it was used for the large-eddy simulation of turbulent flows. Thus, we have a large experience of the SVV stabilization for the computation of turbulent wake flows.

Projection techniques.

A projection technique is used to enforce the divergence free constraint of the current. Projection techniques have been developed for a long time, in the frame of the Navier-Stokes equations to provide efficient algorithms when incompressible flows are concerned. For the Braginskii system, it appears natural to make use of such techniques for the current. Different projection techniques have been implemented in the FBGKI code, from the most classical one (Chorin-Temam, 1969) to the most recent. It turns out however that using projection techniques is less straightforward for Braginskii than for Navier-Stokes. We actively work on this point in order to cure some not yet understood failures of convergence with the time-step.

In the frame of the Eurofusion program, it is planned to check this version of the code on a simple configuration proposed by the EPFL (Paolo Ricci) where experimental as well as numerical results are available.