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In the presented field line map approach the simulation domain of a tokamakis covered with a cylindrical grid, which is Cartesian within poloidal planes.Standard finite-difference methods can be used for the discretisation ofperpendicular (w.r.t.~magnetic field lines) operators. The characteristic flutemode property $\left(k_{\parallel}\ll k_{\perp}\right)$ of structures isexploited computationally by a grid sparsification in the toroidal direction. Afield line following discretisation of parallel operators is then required,which is achieved via a finite difference along magnetic field lines. Thisincludes field line tracing and interpolation or integration. The main emphasisof this paper is on the discretisation of the parallel diffusion operator.Based on the support operator method a scheme is constructed which exhibitsonly very low numerical perpendicular diffusion. The schemes are implemented inthe new code GRILLIX, and extensive benchmarks are presented which show thevalidity of the approach in general and GRILLIX in particular. The mainadvantage of the approach is that it does not rely on field/flux-aligned, whichbecome singular on the separatrix/X-point. Most tokamaks are based on thedivertor concept, and the numerical treatment of the separatrix is therefore ofimportance for simulations of the edge and scrape-off layer.

The field line map approach for simulations of magnetically confined plasmas