Next generation time stepping schemes for weather and climate prediction

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.