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革命，这是由处理器组件的尺寸和速度的物理限制所驱动的。这在我们需要运行的预测模拟和可能在硬件[5]上运行的模拟之间打开了一个“鸿沟”。未来的硬件将由更多但更弱的处理器组成，这意味着我们必须将计算分布在多个处理器之间，这样它们才能同时计算，或者“并行”计算。这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英国气象局的GungHo项目，其动机是与网格几何结构相关的并行通信瓶颈。结果是使用兼容的有限元方法实现新的空间离散化，该方法保留了运动方程的基本属性，而不会对网格几何形状施加限制[1,2]。然而，这并不能解决空间域分解固有的并行可扩展性问题：我们必须找到一种在时域内进行并行计算的方法。虽然时间并行方法听起来违反直觉，因为我们预计大气的未来状态顺序依赖于它的过去状态，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。特别有趣的是平行方法，它使用一种精确的方案来迭代地改进，并行地，计算成本低的“粗传播器”的输出，可以采取大的时间步长。由于大气流动的快速波限制了粗传播子的时间步长，因此以这种方式建模具有挑战性。在b[4]中提出的解决方案是包括近共振波的影响。当应用于理想配置时，该算法证明了显著的并行加速。该项目将继续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，重点关注动力学中没有时间尺度分离的情况。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）来运行，它可以快速构建与气象局直接相关的新方案的原型。进一步的研究取决于学生的兴趣，但可能包括研究非连续物理参数化方案对收敛的影响。这将涉及实现b[3].1中的潮湿浅水模型。亚当斯，萨曼莎V，等人。“LFRic：应对天气和气候模型的可扩展性和性能可移植性的挑战。”并行与分布式计算学报（2019）柯林·J·科特和杰玛·希普顿。数值天气预报的混合有限元。计算物理学报，2001,21(2012):7076-7091.3。弗格森，贾里德·欧，克里斯蒂安·雅布罗诺夫斯基和汉斯·约翰森。“评估带水分的强迫浅水模型的自适应网格细化（AMR）”。3 .《天气月报》（2019）；豪特、特里和贝丝·温盖特。高振荡偏微分方程的渐近时间并行方法。科学计算学报36.2 (2014):A693-A713.5。布莱恩·N·劳伦斯等人。“跨越鸿沟：如何为下一代计算机开发天气和气候模型。”地球科学模型发展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.