| Contents |
| In the context of the ITER magnetic fusion project it is important to simulate the behavior of edge instabilities in the Tokamak. Indeed these instabilities
can damage wall components due to their extremely high energy transfer rate. Consequently, it is essential to estimate the amplitude of these instabilities and understand how control these.
To simulate this phenomena, we use a 3D reduced MHD code in cylindrical geometry named Jorek. The full MHD model is given by
\begin{equation}\label{MHD}
\left\{\begin{array}{l}
\displaystyle\partial_t \rho+\nabla.(\rho\mathbf{v})=0\\
\\
\displaystyle\rho\partial_t \mathbf{v}+\rho\mathbf{v}.\nabla \mathbf{v}+\nabla (\rho T)=\mathbf{J}\times\mathbf{B}+\nu \triangle \mathbf{v}\\
\\
\displaystyle \partial_t p+\mathbf{v}\nabla p+\gamma p\nabla \mathbf{v}=0\\
\\
\displaystyle \partial_t \mathbf{B}=\nabla \times (\mathbf{v}\times\mathbf{B}-\eta\mathbf{J})\\
\\
\displaystyle \nabla.\mathbf{B}=0
\end{array}\right.
\end{equation}
The reduced resistivity MHD models are designed to reduce the CPU cost using the properties of the plasma in the Tokamak configuration. The idea is to write the magnetic and velocity fields using a potential formulation adapted in the tokamak configuration and write the equations on these potentials. At the end the magnetic and velocity fields are given by
$$
\mathbf{B}=\frac{F_0}{R}e_{\phi}+\frac{1}{R}\nabla \psi \times e_{\phi}
$$
and
$$
\mathbf{v}=-R\nabla u \times e_{\phi}+v_{||}\mathbf{B}
$$
Firstly we propose some results about the rigorous derivation of the different reduced models and the conservation of the total energy (important for numerical stability).
Secondly we propose to replace the implicit linear scheme used in Jorek by nonlinear time solvers as a Newton method or a continuation method with adaptive time stepping. We will show that these methods are more robust and efficient to compute correctly the physical instabilities.
To finish we will propose some theoretical and numerical results on a physic preconditioning for the reduced MHD models. This preconditioning is based on a splitting between the diffusion terms and the hyperbolic systems and a parabolization of the hyperbolic coupling operators. We obtain a wave operator for the reduced MHD models which allows to design preconditioning operators. |
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