Near-resonance-informed parallel methods for nonlinear oscillatory PDE models

This project investigates a new class of numerical methods, designed for modern parallel-computing architectures, for solving nonlinear partial differential equations (PDEs) with prominent oscillatory behaviour. Such PDEs arise widely in large-scale fluid simulation, weather prediction, and climate modelling. The work is underpinned by the concept of near resonances, developed in recent theoretical studies by Bin Cheng and collaborators, which analytically reveal how fast oscillations can exert substantial influence on longer-term dynamics through a robust near-resonant coupling mechanism that is both ubiquitous and critical in nonlinear dynamics. This theoretical insight motivates the development of numerical schemes capable of capturing these essential interactions accurately and efficiently.

The primary objective is to develop and rigorously assess the proposed class of near-resonance-informed numerical methods, with particular focus on accuracy, parallel speed-up, and hardware/energy efficiency. The project will design, test and analyse parallel-in-time algorithms that distribute linear sub-problems across CPU cores and reconstruct their nonlinear interactions in a manner consistent with near-resonance theory. Benchmarking and performance evaluation will draw on scaling laws, providing practical guidance on the optimal and sustainable use of HPC resources.

The outcome will be a general numerical framework suited to modern and upcoming computational architectures, with potential for significant impact in fluid dynamics, geophysical modelling, and broader fields involving oscillatory PDEs. Through theoretical analysis, algorithm design, and prototype software development, the PGR will contribute to advancing both mathematical understanding and computational capability in an area of growing scientific importance.

You will need to meet the minimum entry requirements for our Mathematics PhD programme.

Open to any UK or international candidates. Up to 30% of our UKRI funded studentships can be awarded to candidates paying international rate fees. Find out more about eligibility.