Finite difference scheme for solving general 3D convection–diffusion equation Original Research Article

The effect of ergodisation (either by additional coils like in the TEXTOR-DED tokamak or by intrinsic plasma effects like in the W7-X stellarator) defines the need for transport models being able to describe this properly. A prerequisite for this is the concept of local magnetic coordinates allowing a correct discretization with mimimized numerical errors. In the present paper we employ the finite difference discretization method which allows the numerical simulation of energy transport in complex 3D edge geometries (in particular for W7-X) using a custom-tailored unstructured grid in local magnetic coordinates. This grid is generated by field-line tracing to guarantee an exact discretization of the dominant parallel transport. Therefore the radial and parallel fluxes are almost perfectly separated which minimizes the numerical diffusion connected with strong anisotropy of the system (D||/D⊥∼107). In addition, the parallel and radial directions can be treated independently in the numerical method. Along the magnetic field lines we use the standard Patankar concept to discretize the convection–diffusion equation. In order to solve a quasi-isotropic problem in a plane (toroidal cut), we modified the finite volume method to obtain its finite difference representation. The radial fluxes on plasma cross-sections are interpolated using a constrained Delaunay triangulation (keeping the structural information for magnetic surfaces if they exist). The general concept of an upwind scheme is used to discretize the convective terms. The first tests for W7-X geometry were successfully performed.