A fast algebraic multigrid solver and accurate discretization for highly anisotropic heat flux I: open field lines
Golo A. Wimmer, Ben S. Southworth, Thomas J. Gregory, Xian-Zhu Tang
TL;DR
This work tackles the challenge of simulating highly anisotropic heat transport in magnetically confined plasmas by introducing a mixed DG discretization with an auxiliary variable for the parallel heat flux and a fast AIR-based algebraic multigrid solver. The discretization yields accurate resolution of the strongly anisotropic term while maintaining a system structure amenable to efficient block-wise inversion; the solver reorders the 2×2 block system so the advection operators are on the diagonal and uses AIR-AMG for those blocks. Numerical results show the method delivers dramatically higher accuracy (up to ~1000× at κ∥/κ⊥ = 10^9 on open field lines) and significantly faster convergence than diffusion-based AMG approaches, especially in regimes with extreme anisotropy; however, the current approach requires open field lines and is extended in future work to handle closed field lines and sheath boundary conditions. Overall, the combination of a high-fidelity DG-upwind discretization with an AIR-based solver provides robust, scalable performance for realistic, highly anisotropic plasma transport simulations.
Abstract
We present a novel solver technique for the anisotropic heat flux equation, aimed at the high level of anisotropy seen in magnetic confinement fusion plasmas. Such problems pose two major challenges: (i) discretization accuracy and (ii) efficient implicit linear solvers. We simultaneously address each of these challenges by constructing a new finite element discretization with excellent accuracy properties, tailored to a novel solver approach based on algebraic multigrid (AMG) methods designed for advective operators. We pose the problem in a mixed formulation, introducing the directional temperature gradient as an auxiliary variable. The temperature and auxiliary fields are discretized in a scalar discontinuous Galerkin space with upwinding principles used for discretizations of advection. We demonstrate the proposed discretization's superior accuracy over other discretizations of anisotropic heat flux, achieving error $1000\times$ smaller for anisotropy ratio of $10^9$, for $closed$ $field$ $lines$. The block matrix system is reordered and solved in an approach where the two advection operators are inverted using AMG solvers based on approximate ideal restriction (AIR), which is particularly efficient for upwind discontinuous Galerkin discretizations of advection. To ensure that the advection operators are non-singular, in this paper we restrict ourselves to considering open (acyclic) magnetic field lines for the linear solvers. We demonstrate fast convergence of the proposed iterative solver in highly anisotropic regimes where other diffusion-based AMG methods fail.