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