This work develops a novel all-at-once space-time preconditioning approach for resistive magnetohydrodynamics (MHD), with a focus on model problems targeting fusion reactor design. We consider parallel-in-time due to the long time domains required to capture the physics of interest, as well as the complexity of the underlying system and thereby computational cost of long-time integration. To ameliorate this cost by using many processors, we thus develop a novel approach to solving the whole space-time system that is parallelizable in both space and time. We develop a space-time block preconditioning for resistive MHD, following the space-time block preconditioning concept first introduced by Danieli et al. in 2022 for incompressible flow, where an effective preconditioner for classic sequential time-stepping is extended to the space-time setting. The starting point for our derivation is the continuous Schur complement preconditioner by Cyr et al. in 2021, which we proceed to generalise in order to produce, to our knowledge, the first space-time block preconditioning approach for the challenging equations governing incompressible resistive MHD. The numerical results are promising for the model problems of island coalescence and tearing mode, with the overhead computational cost associated with space-time preconditioning versus sequential time-stepping being modest and primarily in the range of 2x-5x, which is low for parallel-in-time schemes in general. Additionally, the scaling results for inner (linear) and outer (nonlinear) iterations are flat in the case of fixed time-step size and only grow very slowly in the case of time-step refinement.

翻译：本文针对电阻磁流体力学（MHD）问题，特别是以聚变反应堆设计为目标模型问题，提出了一种全新的全耦合时空预处理方法。由于捕捉相关物理现象需要长时间域，且底层系统复杂度高导致长时间积分计算成本高昂，我们采用时间并行策略。为通过多处理器降低计算成本，我们开发了一种新型空间-时间均可并行化的全时空系统求解方法。基于Danieli等人2022年针对不可压缩流首次提出的时空块预处理概念，我们将其扩展至电阻MHD领域——将经典时间步进中的高效预处理器延伸至时空框架。推导的起点是Cyr等人2021年提出的连续Schur补预处理器，在此基础上我们对其进行泛化，据我们所知，这是首个针对不可压缩电阻MHD控制方程的时空块预处理方法。针对岛合并与撕裂模模型问题的数值结果令人鼓舞：时空预处理相比时间步进方法的额外计算开销较小，主要控制在2-5倍范围内，这在时间并行方案中属于较低水平。此外，内迭代（线性）与外迭代（非线性）的扩展性在固定时间步长下保持平稳，仅随时间步长细化呈现极缓慢增长。