Advances in the flux-coordinate independent approach

The flux-coordinate independent approach (FCI) offers a promising solution to deal with a separatrix and X-point(s) in diverted tokamaks. Whereas the discretisation of perpendicular operators (with respect to magnetic field) is straight forward, the major complexity lies in the discretisation of parallel operators, for which field line tracing and interpolation is employed. A discrete version for the parallel diffusion operator was proposed in Stegmeir et al. (2016), which maintains the self-adjointness property on the discrete level and exhibits only very low numerical perpendicular diffusion/pollution. However, in situations where the field line map is strongly distorted this scheme revealed its limits. Moreover, the appearance of small scale corrugations deteriorated the convergence order with respect to spatial resolution (Held et al., 2016). In this paper we present an extension to the scheme where the parallel gradient is reformulated via a combination of integration and interpolation. It is shown that the resultant scheme finally combines many good numerical properties, i.e. it is self-adjoint on the discrete level, it has very low numerical perpendicular diffusion, it can cope with strongly distorted maps and exhibits optimal convergence. Another subtle issue in the FCI approach is the treatment of boundary conditions, especially where magnetic field lines intersect with material plates. We present a solution based on ghost points, whose value can be set in a flexible way according to Taylor expansion around the boundary.