Next generation time stepping schemes for weather and climate prediction

准确、及时的天气和气候预报在很大程度上依赖于支撑预报模型的数学和数值算法的设计，以及它们利用超级计算机硬件的效率。超级计算机的设计正在经历一场由处理器组件的尺寸和速度的物理限制所驱动的革命。这在我们需要运行的预测模拟和硬件上可能运行的预测模拟之间打开了一个“鸿沟”[5]。未来的硬件将由更多但功能更弱的处理器组成，这意味着我们必须将计算分布在处理器上，以便它们可以同时或“并行”计算。这就需要对数学和数值算法进行革命性的重新设计。这方面的一个例子是最近的英国气象局GungHo项目，其动机是与网格几何形状相关的并行通信瓶颈。其结果是一个新的空间离散使用兼容的有限元方法，保留基本属性的运动方程，而不施加限制网格几何[1，2]。然而，这并不能解决空间域分解中固有的并行可扩展性问题：我们必须找到一种方法在时域中执行并行计算。虽然时间并行方法听起来违反直觉，因为我们期望大气的未来状态顺序地依赖于其过去的状态，但基于指数积分器的方案提供了更大的时间步长和时间并行计算的潜力。特别令人感兴趣的是parareal方法，它使用一个精确的计划，并行迭代细化的计算成本低的“粗传播”，可以采取大的时间步长的输出。由于快波限制了粗传播算子的时间步长，因此以这种方式建模大气流是具有挑战性的。在[4]中提出的解决方案是包括近共振波的影响。该算法在应用于理想配置时已显示出相当大的并行加速比。本项目将继续Wingate和Shipton的工作，开发1）旋转浅水方程的时间并行积分方案和2）新的测试用例，重点放在动力学中没有时标分离的情况下。最初的模拟将使用Gusto动态核心工具包-一个建立在Firedrake库之上的兼容有限元模型-它可以快速原型化与气象局直接相关的新方案。进一步的研究取决于学生的兴趣，但可能包括调查非连续物理参数化方案对收敛的影响。这将涉及到实施一个潮湿的浅水模式[3]。亚当斯，萨曼莎五，LFRic：迎接天气和气候模型中可扩展性和性能可移植性的挑战。Journal of Parallel and Distributed Computing（2019）Cotter，Colin J.还有杰玛·希普顿“混合有限元数值天气预报。”Journal of Computational Physics 231.21（2012）：7076-7091.3. Ferguson，Jared O.，克里斯蒂安·雅布洛诺夫斯基和汉斯·约翰森“评估强迫浅水模型中的自适应网格细化（AMR）。《2019年天气回顾》（Monthly Weather Review）。Haut，Terry，and Beth Wingate.“一种用于高振荡偏微分方程的渐近时间并行方法。SIAM Journal on Scientific Computing 36.2（2014）：A693-A713.5.劳伦斯，布赖恩N.，跨越鸿沟：如何为下一代计算机开发天气和气候模型。Geoscientific Model Development 11.5（2018）：1799-1821.

Accurate, timely weather and climate forecasting strongly relies on the design of the mathematical and numerical algorithms underpinning the forecast model and the efficiency with which they exploit supercomputer hardware. Supercomputer design is undergoing a revolution driven by physical limitations on the size, and therefore speed, of processor components. This opens a `chasm' between the forecast simulations we need to run and what is possible to run on the hardware [5]. Future hardware will consist of vastly more, but less powerful, processers meaning that we must distribute calculations across the processors so they can be computed simultaneously, or `in parallel'. This requires revolutionary redesign of the mathematical and numerical algorithms. An example of this is the recent UK Met Office GungHo project, motivated by parallel communication bottlenecks related to the geometry of the grid. The outcome was a new spatial discretisation using compatible finite element methods which preserve underlying properties of the equations of motion without imposing restrictions on grid geometry [1, 2]. However, this does not solve the parallel scalability problem inherent in spatial domain decomposition: we must find a way perform parallel calculations in the time domain.While time-parallel methods sound counterintuitive since we expect the future state of the atmosphere to depend sequentially on its past state, schemes based on exponential integrators offer potential for larger timesteps and time-parallel computation. Of particular interest is the parareal method, which uses an accurate scheme to iteratively refine, in parallel, the output of a computationally cheap 'coarse propagator' that can take large timesteps. Atmospheric flows are challenging to model in this way due to fast waves which limit the timestep of the coarse propagator. The solution, proposed in [4], is to include the effects of near resonant waves. This algorithm has demonstrated substantial parallel speedup when applied to idealised configurations.This project will continue the work of Wingate and Shipton in developing 1) time-parallel integration schemes for the rotating shallow water equations and 2) new test cases which focus on the situation where there is no timescale separation in the dynamics. Initially simulations will be run using the Gusto dynamical core toolkit - a compatible finite element model built on top of the Firedrake library - which enables rapid prototyping of new schemes which are directly relevant to the Met Office.Further research depends on the interests of the student but could include investigating the impact of non-continuous physics parameterisation schemes on convergence. This would involve implementing a moist shallow water model as in [3].1. Adams, Samantha V., et al. "LFRic: Meeting the challenges of scalability and performance portability in Weather and Climate models." Journal of Parallel and Distributed Computing (2019).2. Cotter, Colin J., and Jemma Shipton. "Mixed finite elements for numerical weather prediction." Journal of Computational Physics 231.21 (2012): 7076-7091.3. Ferguson, Jared O., Christiane Jablonowski, and Hans Johansen. "Assessing Adaptive Mesh Refinement (AMR) in a Forced Shallow-Water Model with Moisture." Monthly Weather Review (2019).4. Haut, Terry, and Beth Wingate. "An asymptotic parallel-in-time method for highly oscillatory PDEs." SIAM Journal on Scientific Computing 36.2 (2014): A693-A713.5. Lawrence, Bryan N., et al. "Crossing the chasm: how to develop weather and climate models for next generation computers." Geoscientific Model Development 11.5 (2018): 1799-1821.