论文标题
LFRIC大气模型的混合有限元动力学核心的跨部预处理
Multigrid preconditioners for the mixed finite element dynamical core of the LFRic atmospheric model
论文作者
论文摘要
由于地球物理流体动力学中时间尺度的分离很大，半幅度时间积分器通常用于操作大气预测模型中。它们保证了快速（声学和重力）波的稳定处理，同时又不限制时间步长的严重限制。为了在时间前进地传播大气的状态，必须在每个时间步长求解预后变量的非线性方程。由于非线性通常很弱，因此使用少量的牛顿或皮卡德迭代来完成，这又需要在具有$ \ MATHCAL {O}的线性方程式上有效的解决方案（10^6-10^9）$未知。该线性求解通常是该模型中最昂贵的一部分。在本文中，描述了目前由大都会办公室开发的LFRIC下一代模型的有效线性求解器。该模型使用高级模拟有限元离散化，这使得与使用标准有限差分和有限体积方法的模型相比，有效的求解器的构建具有挑战性。线性求解器在Schur-Complement System的定制多机预处理上取决于压力校正。通过与Krylov-Subspace方法相比，在标准测试用例和现实的模型设置中，证明了Multigrid算法的出色性能和鲁棒性。在生产模式下，该模型必须在100,000秒的处理元素上并行运行。正如数值实验所证实的那样，多机求解器的一个特殊优势是由于避免了昂贵的全球减少操作，因此其出色的并行可伸缩性。
Due to the wide separation of time scales in geophysical fluid dynamics, semi-implicit time integrators are commonly used in operational atmospheric forecast models. They guarantee the stable treatment of fast (acoustic and gravity) waves, while not suffering from severe restrictions on the timestep size. To propagate the state of the atmosphere forward in time, a non-linear equation for the prognostic variables has to be solved at every timestep. Since the nonlinearity is typically weak, this is done with a small number of Newton- or Picard- iterations, which in turn require the efficient solution of a large system on linear equations with $\mathcal{O}(10^6-10^9)$ unknowns. This linear solve is often the computationally most costly part of the model. In this paper an efficient linear solver for the LFRic next-generation model, currently developed by the Met Office, is described. The model uses an advanced mimetic finite element discretisation which makes the construction of efficient solvers challenging compared to models using standard finite-difference and finite-volume methods. The linear solver hinges on a bespoke multigrid preconditioner of the Schur-complement system for the pressure correction. By comparing to Krylov-subspace methods, the superior performance and robustness of the multigrid algorithm is demonstrated for standard test cases and realistic model setups. In production mode, the model will have to run in parallel on 100,000s of processing elements. As confirmed by numerical experiments, one particular advantage of the multigrid solver is its excellent parallel scalability due to avoiding expensive global reduction operations.
