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

Abstract

We present a finite element analogue to the second-order, finite difference scheme for the solution of the heat diffusion equation in strongly magnetised plasma given in Ref. [1]. 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 Ref. [1]. 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 0 = r 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

A mixed implicit–explicit finite difference scheme for heat transport in magnetised plasmas

“…In previous papers [1,2] we have outlined several variants of a numerical scheme which minimizes the pollution of the weak heat fluxes perpendicular to magnetic field lines, by the potentially much more efficient transport along them (typical ratio of heat conductivities in the core of a present tokamak experiment > ⊥ χ χ // 10 10 ). In fact this scheme has been successfully applied to realistic situations, like the 3-d problem of heat transport across magnetic islands and in ergodic layers [3].…”

“…In Refs. [1,2] we considered the solution to the stationary problem, arguing that the same treatment of the spatial derivatives could be used in implicit formulations of the time-dependent equation. In fact this has been successfully done in Refs.…”

Local and Nonlocal Parallel Heat Transport in General Magnetic Fields

“…The use of high-order discretizations has been shown to mitigate numerical pollution of the perpendicular dynamics in finite-difference [3,4] and finite-element methods [4,5]. A maximum principle has been shown to be enforceable by the use of limiters at the discrete level in finite differences [6] and finite elements [7], albeit with the effect of rendering both spatial discretizations formally first-order accurate.…”

Numerical approach to the parallel gradient operator in tokamak scrape-off layer turbulence simulations and application to the GBS code

“…However, the FCI method has been tested for closed field line configuration only. Additionally, several magnetohydrodynamic and turbulence studies have been performed without taking advantage of field-aligned property [14][15][16][17]. Although the required resolution in such cases is higher compared to field-aligned methods, the implementation is usually simpler.…”