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 heat flux as an auxiliary variable and discretizing the temperature and auxiliary fields in a discontinuous Galerkin space. The resulting block matrix system is then reordered and solved using an approach in which 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. We demonstrate the proposed discretization's superior accuracy over other discretizations of anisotropic heat flux, achieving error $1000\times$ smaller for anisotropy ratio of $10^9$, while also demonstrating fast convergence of the proposed iterative solver in highly anisotropic regimes where other diffusion-based AMG methods fail.
翻译:我们为厌食性热通量方程式提出了一个新颖的求解器技术,目的是在磁性封闭聚变等离子体中看到高水平的抗生素,这些问题提出了两大挑战:(一) 离散精度和(二) 高效的隐含线求解器。我们同时解决其中的每一项挑战,我们根据一种基于振动操作器的代谢性多格(AMG)方法的新型求解器方法,专门设计出一种以振动操作器为主的新型求解器方法。我们在一个混合的配方中提出问题,将热通量作为一种辅助变量,在不连续的加勒金空间中将温度和辅助字段分解。由此产生的块矩阵系统随后重新排序和解决,采用一种方法,即使用基于近似理想限制(AIR)的AMG解算器(AIR)将两个吸附在倒流中,对于不连续的加勒金多格(AMG)离心器离心器离心器离心法特别有效。在本文中,我们仅限于考虑开放(环流磁场)磁场的磁场线。我们展示了离心 10美元的快速递化高压的离解性硬度,同时展示了另一个的离心率。