Numerical modelling of strongly anisotropic dissipative effects in MHD

In magnetically confined fusion plasmas there is extreme anisotropy due to the high temperature and large magnetic field strength to the extent that thermal conductivity coefficients can be up to $10^{12}$ times larger in the parallel direction than in the perpendicular direction. This anisotropy puts stringent requirements on the numerical methods. A common approach uses field aligned coordinates but in case of magnetic x-points and reconnection local non-alignment is unavoidable. Accuracy in case of high levels of anisotropy for non-field aligned grids is needed for the simulation of instabilities and radial transport processes in the presence of magnetic reconnection, e.g. with edge turbulence. % We therefore consider $2^{nd}$ order numerical schemes which are suitable for non-aligned grids. A novel method for co-located grids, developed to take into account the direction of the magnetic field, has been applied to the unsteady anisotropic heat diffusion equation on a non-field-aligned grid and compared with several other discretisation schemes, including G{\"{u}}nter et al's symmetric scheme. Test cases include variable diffusion coefficients with anisotropy values up to $10^{12}$, and field line bending in divergence and non-divergence free (unit vector) field configurations. % One of the model problems is given by the unsteady heat equation % \begin{equation*} \begin{split} \mbf{q} &= - D_\bot\nabla T - (D_\|-D_\bot)\mbf{b}\mbf{b}\cdot\nabla T, \quad \diff{T}{t} = -\nabla\cdot\mbf{q} + f, \end{split} \label{eq:braginskii} \end{equation*} where $T$ represents the temperature, $\mbf{b}$ represents the unit direction vector of the magnetic field line with respect to the coordinate axes, $f$ is some source term and $D_\|$ and $D_\bot$ represent the parallel and the perpendicular diffusion coefficient respectively. \\ % Preliminary conclusions are that for FDM's the preservation of self-adjointness is crucial for limiting the pollution of perpendicular diffusion to acceptable values. However it is not required for maintaining the order of accuracy in most cases as is demonstrated by our aligned method. Key goal is to improve the co-located method to obtain acceptable levels for the pollution of the perpendicular diffusion whilst maintaining convergence.