We propose an optimally performant fully implicit algorithm for the Hall magnetohydrodynamics (HMHD) equations based on multigrid-preconditioned Jacobian-free Newton-Krylov methods. HMHD is a challenging system to solve numerically because it supports stiff fast dispersive waves. The preconditioner is formulated using an operator-split approximate block factorization (Schur complement), informed by physics insight. We use a vector-potential formulation (instead of a magnetic field one) to allow a clean segregation of the problematic $\nabla \times \nabla \times$ operator in the electron Ohm's law subsystem. This segregation allows the formulation of an effective damped block-Jacobi smoother for multigrid. We demonstrate by analysis that our proposed block-Jacobi iteration is convergent and has the smoothing property. The resulting HMHD solver is verified linearly with wave propagation examples, and nonlinearly with the GEM challenge reconnection problem by comparison against another HMHD code. We demonstrate the excellent algorithmic and parallel performance of the algorithm up to 16384 MPI tasks in two dimensions.

翻译：我们提出了一种基于多重网格预处理雅可比自由牛顿-克里洛夫方法的、性能最优的全隐式算法，用于求解霍尔磁流体动力学（HMHD）方程组。HMHD是一个数值求解极具挑战性的系统，因为它支持刚性的快速色散波。该预处理器利用基于物理见解的算子分裂近似块分解（舒尔补）进行构建。我们采用矢量势形式（而非磁场形式），以便在电子欧姆定律子系统中清晰地分离出棘手的 $\nabla \times \nabla \times$ 算子。这种分离使得我们能够为多重网格构建一个有效的阻尼块雅可比光滑子。我们通过分析证明，所提出的块雅可比迭代是收敛的并具有光滑性质。通过波传播算例对所得HMHD求解器进行了线性验证，并通过与另一个HMHD代码的对比，利用GEM挑战重联问题进行了非线性验证。我们在二维情况下展示了该算法在多达16384个MPI任务上的优异算法性能和并行性能。