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，这场革命是由处理器组件的尺寸和速度的物理限制推动的。这打开了我们需要运行的预测模拟与硬件上可能运行的模拟之间的“鸿沟”[5]。未来的硬件将由更多但能力更弱的处理器组成，这意味着我们必须将计算分布在各个处理器上，以便它们可以同时计算，也就是“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。最近的英国气象局GungHo项目就是一个例子，该项目的动机是与电网几何结构相关的并行通信瓶颈。其结果是使用兼容的有限元方法进行新的空间离散，该方法保留了运动方程的基本性质，而不对网格几何施加限制[1，2]。然而，这并没有解决空间域分解固有的并行可伸缩性问题：我们必须找到在时间域执行并行计算的方法。虽然时间并行方法听起来有悖于直觉，因为我们预计大气的未来状态将顺序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可能性。特别令人感兴趣的是Parareal方法，它使用一种精确的方案来迭代地并行地精炼计算成本较低的可能需要很大时间步长的“粗传播子”的输出。由于快速波限制了粗传播子的时间步长，以这种方式对大气流动进行建模是具有挑战性的。文[4]中提出的解决方案是考虑近谐振波的影响。这个项目将继续Wingate和Shipton的工作，为旋转浅水方程开发1)时间并行积分格式，2)新的测试案例，专注于动力学中没有时间尺度分离的情况。最初，模拟将使用Gusto Dynamic core工具包--建立在Firedrake库之上的兼容有限元模型--来实现与气象局直接相关的新方案的快速原型。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。这将涉及实施如[3].1中所述的潮湿浅水模型。亚当斯、萨曼莎·V等人。“LFric：在天气和气候模型中应对可伸缩性和性能可移植性方面的挑战。”《并行与分布式计算学报》(2019)。题名/责任者：A.“数值天气预报的混合有限元。”计算物理杂志231.21(2012年)：第7076-7091.3页。弗格森、贾里德·O、克里斯蒂安·贾布洛诺夫斯基和汉斯·约翰森。在有水分的强迫浅水模式中评估自适应网格细化(AMR)。《天气月报》(2019)。豪特、特里和贝丝·温盖特。“高振荡偏微分方程解的渐近并行时间方法。”暹罗科学计算期刊36.2(2014)：A693-A713.5。Lawrence，Bryan N.，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.