EPSRC Reference: 
EP/R029628/1 
Title: 
On the way to the asymptotic limit: mathematics of slowfast coupling in PDEs 
Principal Investigator: 
Wingate, Professor B 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics 
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: 

Related Grants: 

Panel History: 

Summary on Grant Application Form 
The main motivation for this proposal is the pressing need to understand oscillatory stiffness with finite timescale 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 nextgeneration 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 finitetime 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 nearresonantsets 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 frequencyaveraging developed by the PI and her collaborators.
The finite timescale 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 nearresonances completely missed by implicit methods, and construct the first proofs of superlinear convergence for oscillatory timeparallel numerical methods. In doing so, their algorithm achieved parallel speedups of over 100 over standard methods.
Given the crucial role of oscillations in reducing timetosolution for the numerical solution of PDEs, the time is right for advancing the subject. This project proposes a solution to address this longstanding problem of finite timescale separation in PDEs by uniting PDEs analysis, numerical analysis, continuum dynamics, and computational science. It provides a direct lineofsight from the heart of mathematical analysis into advances required to meet the goals for nextgeneration high performance computing.

Key Findings 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Potential use in nonacademic contexts 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Impacts 
Description 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk 
Summary 

Date Materialised 


Sectors submitted by the Researcher 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Project URL: 

Further Information: 

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