Abstract:

The anisotropic diffusion equation is used in fusion plasma physics research for a number of purposes; for instance as a simplified model for the diffusion of particles, to model the temperature or pressure in fusion device, or as a basis for investigating the magnetic field structures hiding in chaotic layers. In fusion plasmas, the strength of the magnetic field results in the transport along field lines being orders of magnitude faster than across them. This can result in the ratio of parallel to perpendicular diffusion coefficients exceeding $\kappa_\parallel/\kappa_\perp>10^{8}$. The extreme anisotropy manifests in an ill conditioned problem which presents computational challenges.

We present a provably stable method for solving the anisotropic diffusion equation. Operator splitting is used to separate the fast and slow scales of the diffusion. For the perpendicular diffusion, we discretise using the Summation By Parts approach. The parallel component is then added using a penalty which is constructed by field line tracing. The full discretisation can be proved to be unconditionally stable. Convergence results are shown for a manufactured solution and results are shown for example with a single magnetic island. Finally, some new results are shown for a circular domain.