Phase Averaged Deferred Correction for Multi-Timescale Systems

The time-dependent systems are described with differential equations (either ordinary or partial differential equations). These systems are everywhere from ocean and atmospheric flows to financial markets or biological models. To know their behaviour, we need to solve the differential equations by integrating them in time. We often do this numerically using our computational resources. Such computation for complicated systems like weather models is very demanding and takes a long time. To lower the computational time, in modern scientific computing we parallelise the computational tasks and assign them to different processors that compute them simultaneously. However, there is a big obstacle in the parallisation of time integration for solving differential equations. Time integration is a sequential process, in which computing the solution at any timestep requires the solution at previous timesteps. Hence, it cannot be parallelised easily. The efficient time integration of nonlinear multi-timescale systems poses an additional challenge. The fast modes of these systems are coupled with the slow modes and finding their solution requires very small timesteps that slow down the overall computation. This project addresses these two challenges (parallelisation of time integration and fast oscillations) by developing a novel parallel time integrator that efficiently computes the solution of nonlinear multi-timescale systems.

The method that we plan to develop considers the differential equations averaged over the phase of fast oscillations. The averaging is done in a systematic way such that it will be easy to retrieve the fast dynamics from the averaged solution. The biggest advantage of this averaging is to allow taking larger times without compromising too much on the accuracy or the solution blowing up due to numerical instabilities. The averaging itself, however, introduces a new type of error in computation. To mitigate this effect, we iteratively correct the averaged solution by lowering the averaging window.

A part of our method's novelty is designing these correction layers in a way that can be computed in parallel and hence using several processors to lower to the overall computation time. After developing our method and testing it on simple examples, we apply it to a model of shallow waters that incorporates fast waves and slow vortices. This can be a stepping-stone for the application of the proposed method in more complicated geophysical flows in the ocean and weather prediction models.