A team lead by Beth Wingate (Exeter) and including Bin Cheng (Surrey), Colin Cotter (Imperial), and David Acreman (Exeter) has been awarded an £850k EPSRC grant for the project “On the way to the asymptotic limit: mathematics of slow-fast coupling in PDEs” (EPSRC link here). The grant runs till March 2021.
The main motivation of the project is the pressing need to understand oscillatory stiffness with finite time-scale separation in PDEs and its effects on their numerical analysis. Since the beginnings of numerical weather prediction, a key challenge has been the intimate relationship between the mathematical structure of the equations, their numerical approximation, and the associated physical phenomenon.
This project proposes that a fruitful path to advancing the understanding of oscillations in nonlinear PDEs is the theory of finite-time scale separation, which is fundamentally an issue of 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 is a major challenge and is one of the themes that will be explored in this project through a novel technique of frequency-averaging developed by the PI and her collaborators.