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Details of Grant 

EPSRC Reference: EP/R029628/1
Title: On the way to the asymptotic limit: mathematics of slow-fast coupling in PDEs
Principal Investigator: Wingate, Professor B
Other Investigators:
CHENG, Dr B Cotter, Dr C Acreman, Dr DM
Researcher Co-Investigators:
Dr P Burns
Project Partners:
Department: Mathematical Sciences
Organisation: University of Exeter
Scheme: Standard Research
Starts: 01 August 2018 Ends: 31 July 2022 Value (£): 849,609
EPSRC Research Topic Classifications:
Computer Sys. & Architecture Continuum Mechanics
Mathematical Analysis Numerical Analysis
Parallel Computing
EPSRC Industrial Sector Classifications:
Information Technologies
Related Grants:
Panel History:
Panel DatePanel NameOutcome
27 Feb 2018 EPSRC Mathematical Sciences Prioritisation Panel February 2018 Announced
Summary on Grant Application Form
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.

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Organisation Website: http://www.ex.ac.uk