Time parallel integration and phase averaging for the nonlinear shallow water equations on the sphere

TL;DR

This paper introduces a phase averaging technique for the nonlinear shallow water equations on the sphere, enabling larger timesteps by capturing slow dynamics, with a focus on stability, accuracy, and parallel implementation.

Contribution

It develops a stable matrix exponential for finite elements and a parallel finite averaging procedure, advancing phase averaging methods for geophysical fluid dynamics.

Findings

Optimal averaging window identified, matching theoretical predictions.

Combined discretisation and averaging errors are smaller than traditional semi-implicit methods.

Parallel averaging integral evaluated efficiently using Riemann sums.

Abstract

We describe a proof-of-concept development and application of a phase averaging technique to the nonlinear rotating shallow water equations on the sphere, discretised using compatible finite element methods. Phase averaging consists of averaging the nonlinearity over phase shifts in the exponential of the linear wave operator. Phase averaging aims to capture the slow dynamics in a solution that is smoother in time (in transformed variables) so that larger timesteps may be taken. We overcome the two key technical challenges that stand in the way of studying the phase averaging and advancing its implementation: 1) we have developed a stable matrix exponential specific to finite elements and 2) we have developed a parallel finite averaging proceedure. Following Peddle et al (2019), we consider finite width phase averaging windows, since the equations have a finite timescale separation. In our numerical implementation, the averaging integral is replaced by a Riemann sum, where each term can be evaluated in parallel. This creates an opportunity for parallelism in the timestepping method, which we use here to compute our solutions. Here, we focus on the stability and accuracy of the numerical solution. We confirm there is an optimal averaging window, in agreement with theory. Critically, we observe that the combined time discretisation and averaging error is much smaller than the time discretisation error in a semi-implicit method applied to the same spatial discretisation. An evaluation of the parallel aspects will follow in later work.