Spectral methods for partial differential equations on irregular domains

Overview

Spectral methods are among the most efficient methods for solving many important partial differential equations on regular domains (like a rectangle). However, when the problem domain is not a regular shape there are various ways in which the approach can be adapted, each with their own strengths and weaknesses. This project will explore a semi-novel approach that leads to a very elegant formulation, with ease-of-use for the end user being paramount.

Research engagement

Literature review

Mathematical analysis

Algorithm development and implementation

Research activities

Derive algorithms for spectral methods adapted to particular geometries

Extend existing algorithms to new scenarios including unsteady problems, nonlinear problems, eigenvalue problems

Contributing new algorithms and components to an existing software library for spectral methods

Outcomes

Contribution to open source software for solving PDEs using spectral methods

Skills and experience

Coding experience in Julia, Python, MATLAB, or similar is required.  Julia is the language that will be used.

Mathematical training covering partial differential equations (e.g. MXB322) and advanced linear algebra (e.g. MXB201) is essential.

Start date

1 November, 2024

End date

1 February, 2025

Location

QUT Gardens Point

Keywords

• spectral methods
• partial differential equations
• open source
• julia
• numerical methods
• computational mathematics

Contact

Tim Moroney  t.moroney@qut.edu.au