Mathematical models involving partial differential equations with fine-scale variation in coefficients are ubiquitous in engineering and scientific disciplines. For such problems, direct numerical solution of the governing equations via spatial discretisation yields a prohibitively large number of mesh elements and hence a discrete system which is exorbitantly expensive to solve. Such numerical issues motivate the need for spatial-averaged models that aim to efficiently approximate the averaged fine-scale solution on a much coarser mesh. This project will investigate spatially-averaged models for transport processes such as diffusion in heterogeneous domains.
Relevant Literature: EJ Carr, IW Turner and P Perré (2017) Macroscale modelling of multilayer diffusion: using volume averaging to correct the boundary conditions, Applied Mathematical Modelling, 47, p. 600-618. More information can be found on Dr Carr's personal research website.
You will meet regularly with Dr Carr to acquire new knowledge, brainstorm ideas, discuss your progress and receive direction on future work. This project will involve developing mathematical models, implementing code in MATLAB and communicating your work in written and oral forms.
You will develop new skills in numerical computation (MATLAB), mathematical typesetting (LaTeX) and the solution and analysis of partial differential equations. Masters and PhD students will be expected to publish their work in scientific/mathematical journals. For prospective VRES and Honours students, there is also potential for you to contribute new results to the literature and publish your work in the form of a journal article.
Skills and experience
This project can be tailored to either an undergraduate (VRES), Honours, Masters or PhD student and can be personalised to suit your individual interests and skills. Some proficiency in MATLAB, computational mathematics and differential equations is assumed.
- spatial averaging
- partial differential equations
- applied and computational mathematics
Contact the supervisor for more information.