We propose a second-order temporally implicit, fourth-order-accurate spatial discretization scheme for the strongly anisotropic heat transport equation characteristic of hot, fusion-grade plasmas. Following [Du Toit et al., Comp. Phys. Comm., 228 (2018)], the scheme transforms mixed-derivative diffusion fluxes (which are responsible for the lack of a discrete maximum principle) into nonlinear advective fluxes, amenable to nonlinear-solver-friendly monotonicity-preserving limiters. The scheme enables accurate multi-dimensional heat transport simulations with up to seven orders of magnitude of heat-transport-coefficient anisotropies with low cross-field numerical error pollution and excellent algorithmic performance, with the number of linear iterations scaling very weakly with grid resolution and grid anisotropy, and scaling with the square-root of the implicit timestep. We propose a multigrid preconditioning strategy based on a second-order-accurate approximation that renders the scheme efficient and scalable under grid refinement. Several numerical tests are presented that display the expected spatial convergence rates and strong algorithmic performance, including fully nonlinear magnetohydrodynamics simulations of kink instabilities in a Bennett pinch in 2D helical geometry and of ITER in 3D toroidal geometry.

翻译：本文针对高温聚变级等离子体特有的强各向异性热输运方程，提出了一种时间二阶隐式、空间四阶精度的离散化方案。该方案遵循[Du Toit等人，《Comp. Phys. Comm.》，228卷（2018年）]的思路，将混合导数扩散通量（其导致离散极值原理失效）转化为非线性对流通量，从而可应用对非线性求解器友好的保单调限制器。该方案能够以低跨场数值误差污染和优异的算法性能，精确模拟热输运系数各向异性高达七个数量级的多维热输运问题，其线性迭代次数随网格分辨率和网格各向异性的增长非常微弱，且随隐式时间步长的平方根增长。我们提出了一种基于二阶精度近似的多重网格预处理策略，使该方案在网格细化下保持高效性和可扩展性。文中给出了若干数值测试，展示了预期的空间收敛率和强大的算法性能，包括在二维螺旋几何中Bennett箍缩的扭曲不稳定性以及三维环面几何中ITER的完全非线性磁流体动力学模拟。