Parallel-in-time multi-level integration of the shallow-water equations on the rotating sphere

Abstract

The modeling of atmospheric processes in the context of weather and climate simulations is an important and computationally expensive challenge. The temporal integration of the underlying PDEs requires a very large number of time steps, even when the terms accounting for the propagation of fast atmospheric waves are treated implicitly. Therefore, the use of parallel-in-time integration schemes to reduce the time-to-solution is of increasing interest, particularly in the numerical weather forecasting field.We present a multi-level parallel-in-time integration method combining the Parallel Full Approximation Scheme in Space and Time (PFASST) with a spatial discretization based on Spherical Harmonics (SH). The iterative algorithm computes multiple time steps concurrently by interweaving parallel highorder fine corrections and serial corrections performed on a coarsened problem. To do that, we design a methodology relying on the spectral basis of the SH to coarsen and interpolate the problem in space.The methods are evaluated on the shallow-water equations on the sphere using a set of tests commonly used in the atmospheric flow community. We assess the convergence of PFASST-SH upon refinement in time. We also investigate the impact of the coarsening strategy on the accuracy of the scheme, and specifically on its ability to capture the high-frequency modes accumulating in the solution. Finally, we study the computational cost of PFASST-SH to demonstrate that our scheme resolves the main features of the solution multiple times faster than the serial schemes.

Perspectives of Physics-Based Machine Learning for Geoscientific Applications Governed by Partial Differential Equations

“…Similarly, parallelization in time can strongly relax the fundamental constrain of one time step depending on the previous one (Bauer et al, 2021;Carraro et al, 2015;Fisher and Gürol, 2017;Hamon et al, 2020). Interestingly, some physics-based machine learning techniques provide parallel-in-time schemes .…”

“…A parareal implementation for PINNs, called PPINNs has been already developed in . Similarly, the Parallel Full Approximation Scheme in Space and Time (PFASST) Minion, 2012, 2014;Hamon et al, 2020) falls into the same class as parareal algorithm.…”

GParareal: a time-parallel ODE solver using Gaussian process emulation

“…Runtime issues also arise when solving IVPs with spatial or other nontemporal dependencies in that, even though highly efficient domain decomposition methods exist (Dolean et al 2015), the parallel speed-up of such methods on high performance computers (HPCs) is still constrained by the serial nature of the time-stepping scheme. Therefore, with the advent of exascale HPCs on the horizon (Mann 2020), there has been renewed interest in developing more efficient and robust timeparallel algorithms to reduce wallclock runtimes for IVP simulations in applications spanning numerical weather prediction (Hamon et al 2020), kinematic dynamo modelling (Clarke et al 2020), and plasma physics (Samaddar et al 2010(Samaddar et al , 2019 to name but a few. In this work, we focus on the development of such a time-parallel method.…”