On the way to the asymptotic limit: mathematics of slow-fast coupling in PDEs

The main motivation for this proposal is the pressing need to understand oscillatory stiffness with finite time-scale separation in PDEs and its effects on their numerical analysis. Though this subject has been of interest for many years, because of its wide impact on physical applications, it has become even more important recently because of its potential to revolutionize numerical methods in advance of the shift to next-generation computer architectures (e.g. contenders for ARCHER2 and beyond).

The intimate relationship between the mathematical structure of the equations, their numerical approximation, and the associated physical phenomenon has been understood since the beginning of scientific computing. A classic example is Jules Charney, who, in his 1948 paper, examined weather maps and deduced 'slow' equations that filtered out the fast oscillatory waves responsible for the numerical instabilities first observed by L.F. Richardson in 1922. These equations not only lead to the first realistic numerical weather prediction, but also gave rise to fundamental understanding of the physics of baroclinic instability, the key energy transfer mechanism behind the fluid dynamics of the formation of weather systems.

Since those early days, mathematical analysis, numerical analysts and fluid and continuum dynamics have all made advances toward understanding this subject of oscillations in nonlinear phenomenon. In fact, there are pieces of the puzzle scattered throughout the literature of each subject, but no comprehensive theory has been fully realized.

We propose that a fruitful path to advance the understanding of oscillations in nonlinear PDEs is the missing theory of finite-time scale separation, which is fundamentally an issue of nonlinear interactions, called resonances, between the key frequencies of the PDE. Exact sets of resonances are formed of nonlinear triads and are the only part of the solution that manifests in the asymptotic limit. In physical reality, where the time scale separation is finite, there are additional 'near resonances'. The mathematical definition of near-resonant-sets and their impact on understanding fluid dynamics and advancing mathematical and numerical analysis is a rich ground to cover and is one of the topics that will be explored in this project through a novel technique of frequency-averaging developed by the PI and her collaborators.

The finite time-scale separation case is fundamentally important for advancing numerical methods for next generation computer architectures. By applying this new technique to numerical algorithms, the PI and her collaborators were able to resolve the near-resonances completely missed by implicit methods, and construct the first proofs of superlinear convergence for oscillatory time-parallel numerical methods. In doing so, their algorithm achieved parallel speed-ups of over 100 over standard methods.

Given the crucial role of oscillations in reducing time-to-solution for the numerical solution of PDEs, the time is right for advancing the subject. This project proposes a solution to address this long-standing problem of finite time-scale separation in PDEs by uniting PDEs analysis, numerical analysis, continuum dynamics, and computational science. It provides a direct line-of-sight from the heart of mathematical analysis into advances required to meet the goals for next-generation high performance computing.

The proposed research uses a recently successful theory for frequency averaging in nonlinearity to unite mathematical analysis, fluid mechanics, numerical analysis, and computing to provide new algorithms favourable for application on next-generation computers.

We expect there to be intradisciplinary beneficiaries in each subject area because of the wide reach of the strategically selected research team. But in addition, this project offers a unique opportunity for subjects at the heart of mathematics, like analysis, to have a clear line of sight to advance computing on new architectures. Thus, this project has the potential to accelerate UK mathematics contributions, beyond journal articles, to the global conversation taking place now about next generation computing.

The new wave/mean-flow theory we propose not only benefits fundamental research in fluid mechanics, but we also will apply it to in situ data in the Arctic Ocean, allowing the advances of this research to be accessible to oceanographers and atmosphere and ocean modellers through direct collaboration.

One of the key ideas of this proposal has already demonstrated that it is possible to make significant gains in the parallel performance for computing by reaching deeply into mathematical analysis. By advancing this idea further, the fundamental gains in understanding proposed in this project will continue to benefit numerical analysis and so on into numerical algorithm development. This benefit is likely to have significant impacts in the climate and weather community through already established collaborations with UK Met Office's next-generation numerics, the Gung-Ho project, which the PI and 2 of the Co-I are already members.

Because our new theories will be realised in open-source code in the Firedrake/Gusto framework, it will be of benefit all those who use and develop finite elements.

In addition, one of the Co-I on our project is a Research Software Engineer (RSE) with access to UK's new ARM-based architecture; GW4's Isambard. This new architecture has an international community of its own, making it likely that the advances from this project will be felt in other fields of computing that participate in the ARM architecture. In addition, the advances of this project are likely to be felt in the UK's RSE network (through the annual conference), and therefore, likely to advance UK's national computing capability.