Multigrid preconditioners for the mixed finite element dynamical core of the LFRic atmospheric model
Multigrid preconditioners for the mixed finite element dynamical core of the LFRic atmospheric model
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DOI：
10.1002/qj.3880
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发表时间：
2020-01
影响因子：
8.9
通讯作者：
C. Maynard;T. Melvin;E. Müller
中科院分区：
文献类型：
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作者：
C. Maynard;T. Melvin;E. Müller
由于地球物理流体动力学中时间尺度的广泛分离，半隐式时间积分器通常用于业务大气预报模式。它们保证了快速（声波和重力）波的稳定处理，同时不会受到时间步长的严格限制。为了在时间上向前传播大气的状态，必须在每个时间步求解预报变量的非线性方程。由于非线性通常很弱，这是用少量的牛顿或皮卡德迭代来完成的，这反过来又需要有效地求解一个具有106−109个未知数的大型线性方程组。𝒪这种线性求解通常是模型中计算成本最高的部分。在这篇文章中，描述了一个高效的线性求解器，用于气象局目前正在开发的LFRic下一代模型。该模型使用先进的模拟有限元离散化，与使用标准有限差分和有限体积法的模型相比，这使得构建高效求解器具有挑战性。线性求解器依赖于舒尔补系统的定制多重网格预处理器进行压力校正。通过将其与Krylov子空间方法进行比较，多重网格算法的上级性能和鲁棒性在标准测试用例和实际模型设置中得到了证明。在生产模式下，模型必须在数十万个处理元素上并行运行。数值实验证实，多重网格求解器的一个特别的优点是其良好的并行可扩展性，由于它避免了昂贵的全球减少操作。
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 time‐step size. To propagate the state of the atmosphere forward in time, a nonlinear equation for the prognostic variables has to be solved at every time step. 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 of linear equations with 𝒪(106−109) unknowns. This linear solve is often the computationally most costly part of the model. In this article an efficient linear solver for the LFRic next‐generation model currently being developed by the Met Office is described. The model uses an advanced mimetic finite element discretisation which makes the construction of efficient solvers challenging as 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 it to Krylov subspace methods, the superior performance and robustness of the multigrid algorithm is demonstrated for standard test cases and realistic model set‐ups. In production mode, the model will have to run in parallel on hundreds of thousands of processing elements. As confirmed by numerical experiments, one particular advantage of the multigrid solver is its excellent parallel scalability due to its avoidance of expensive global reduction operations.