The field line map approach for simulations of magnetically confined plasmas

Predictions of plasma parameters in the edge and scrape-off layer of tokamaks is difficult since most modern tokamaks have a divertor and the associated separatrix causes the usually employed field/flux-aligned coordinates to become singular on the separatrix/X-point. The presented field line map approach avoids such problems as it is based on a cylindrical grid: standard finite-difference methods can be used for the discretisation of perpendicular (w.r.t. magnetic field) operators, and the characteristic flute mode property (k∥≪k⊥) of structures is exploited computationally via a field line following discretisation of parallel operators which leads to grid sparsification in the toroidal direction. This paper is devoted to the discretisation of the parallel diffusion operator (the approach taken is very similar to the flux-coordinate independent (FCI) approach which has already been adopted to a hyperbolic problem (Ottaviani, 2011; Hariri, 2013)). Based on the support operator method, schemes are derived which maintain the self-adjointness property of the parallel diffusion operator on the discrete level. These methods have very low numerical perpendicular diffusion compared to a naive discretisation which is a critical issue since magnetically confined plasmas exhibit a very strong anisotropy. Two different versions of the discrete parallel diffusion operator are derived: the first is based on interpolation where the order of interpolation and therefore the numerical diffusion is adjustable; the second is based on integration and is advantageous in cases where the field line map is strongly distorted. The schemes are implemented in the new code GRILLIX, and extensive benchmarks and numerous examples are presented which show the validity of the approach in general and GRILLIX in particular.