Parallel-in-time solution of finite element atmospheric models using ParaDiag

Modern numerical weather prediction requires vast amounts of computing power, so highly scalable algorithms are essential on massively parallel modern hardware. However, once the strong scaling limit is reached for spatial parallelism, the wallclock time increases with the number of timesteps. This limits the ability to provide higher resolution forecasts on an operational schedule. Parallel-in-time methods overcome this limit by exposing time parallelism, in addition to the spatial parallelism exposed by traditional domain decomposition.

ParaDiag is one such method, which reduces the coupled system for multiple time-steps into a block-diagonal matrix that can be solved in parallel. We will present recent progress on the application of ParaDiag to compatible finite element discretisations of PDEs for atmospheric flow, which are particularly challenging for time-parallel methods due to their highly oscillatory nature. These solvers are implemented as an open source general library using Firedrake, an automated code generation framework for the solution of finite element methods.

Various ParaDiag formulations are explored for linear and non-linear models, and their parallel scaling is compared. Their performance is compared against time serial methods to find the parameter ranges where the greatest speedups are achieved. We identify the main difficulties faced by ParaDiag and describe our approaches to overcoming these, including solution strategies for the block systems within the ParaDiag matrix, and improving the convergence of nonlinear models.