Finite element and higher order difference formulations for modelling heat transport in magnetised plasmas

We present a finite element analogue to the second-order, finite difference scheme for the solution of the heat diffusion equation in strongly magnetised plasmas given in Günter et al. [S. Günter et al., J. Comp. Phys. 209 (2005) 354]. Compared to standard finite element or finite difference formulations it strongly reduces the pollution of perpendicular heat fluxes by parallel ones even without resorting to field-aligned coordinates. We present both bi-linear and bi-quadratic versions of this scheme as well as a fourth-order extension of the original difference scheme of Günter et al. (2005). In the second part of the paper, we address the formulation of the boundary conditions at walls with an oblique incidence of field lines and the treatment of the coordinate singularity at r = 0 in cylindrical, or topologically equivalent coordinates with the reduced-pollution finite difference scheme. All tests shown indicate that both the finite-difference and the finite-element versions of the scheme should substantially alleviate the requirement for field-alignment of the coordinates over the realistic range of χ∥/χ⊥ in toroidal magnetic confinement devices.