4 Year GTA - A sparse spectral element method on the sphere for numerical weather prediction

About the Project

Open to UK Applicants only

Mathematics are offering 3 fully-funded Graduate Teaching Assistant (GTA) PhD studentships available for UK applicants, starting in September 2026.

Graduate Teaching Assistantships allow research students to fund their PhD through part-time teaching work with the University.

A Graduate Teaching Assistant is responsible to the Head of School and is expected to undertake teaching or other duties within the School - not normally exceeding 8-10 contact hours per week - while undertaking research leading to a PhD.

Approximately 80% of their time will be spent on their doctoral research and 20% on their GTA responsibilities. Training is provided to help Graduate Teaching Assistants develop their teaching related skills and enhance their professional competencies.

Project Highlights

• The development of the first algorithms to compute vector orthogonal polynomials on subsets of the sphere with optimal complexity
• Combining multiple subsets of the sphere to design a novel spectral element method on the whole sphere
• The application of this method to numerical weather prediction, allowing much higher-resolution simulations than was possible before

Project Description

Spectral methods are used to approximate the solutions to differential equations. Typically, this is done by expansions in orthogonal basis functions on the entire domain of the problem, or on multiple subdomains in a spectral element method.

In recent years, spectral methods have been devised that result in sparse and well-conditioned linear systems that can be solved with optimal complexity algorithms. Furthermore, these sparse spectral methods have been extended from intervals to certain regions in 2D and 3D, for example triangles, balls, cones, disks, disk slices, trapeziums and spherical caps.

The spectral method used by the European Centre for Medium-range Weather Forecasts (ECMWF) uses spherical harmonics (SHs) and vector spherical harmonics (VSHs) as basis functions. The SHs are scalar orthogonal polynomials (OPs) and the VSHs are vector-valued OPs defined on the whole sphere. Since the basis functions are global, the transforms that are required to numerically solve partial differential equations (PDEs) on the sphere can become prohibitively expensive as the degrees of the SHs and VSHs are increased.

The aim of this project is to overcome the parallel scalability bottleneck of this global spectral method by constructing and implementing a sparse spectral element method in which PDEs on the whole sphere are solved by using high-degree scalar and vector-valued OPs on subsets of the sphere such as spherical caps, bands, rectangles and triangles. Since local instead of global basis functions are used, the transforms can be computed much more efficiently. This will allow the use of much higher degree basis functions and therefore higher-resolution simulations of PDEs on the sphere, enabling better resolution of features such as turbulence, with applications not only in numerical weather prediction, but also in astrophysics and geophysics.

Project enquiries to Dr Marco Fasondini m.fasondini@leicester.ac.uk

Application enquiries to pgrapply@le.ac.uk

To apply please refer to the application advice and use the application link at https://le.ac.uk/study/research-degrees/funded-opportunities/maths-gta

Start 21 September 2026

Funding Notes

The 4 year GTA funded studentships provide:

• A combined teaching and stipend payment, currently. for 2026/7 this will be £21,805 per year, paid in monthly instalments
• Tuition fees at UK rates