Many physical processes can be modelled using time and space dependent partial differential equations. The solution of these equations provides researchers/industry with valuable insight into the underlying process and can often be used to explain phenomena that has been observed experimentally. However, due to the complexity of these processes and the irregular geometries on which they apply, exact analytical solutions to the governing equations are almost always not accessible. In such cases, numerical solution methods must be called upon to provide an approximate solution. Typically, the first step is to perform a spatial discretisation of the governing equations over the geometry, which produces an initial value problem involving a large system of ordinary differential equations. In recent times, exponential integrators have emerged as an attractive method for solving such initial value problems. In contrast to classical implicit time-stepping schemes (e.g. backward Euler), which require the solution of a nonlinear system of equations at each time step, exponential integrators are explicit schemes that require the computation of matrix functions involving the matrix exponential. This project will involve developing efficient exponential integration schemes, and related numerical linear algebra techniques, for solving several classical partial differential equations that arise frequently in applied mathematics.
Relevant Literature: EJ Carr, TJ Moroney and IW Turner (2011) Efficient simulation of unsaturated flow using exponential time integration, Applied Mathematics and Computation, 30(14), p. 6587-6596.
You will meet regularly with the supervisory team to acquire new knowledge, brainstorm ideas, discuss your progress and receive direction on future work. This project will involve deriving new mathematical techniques, developing code in MATLAB and communicating your work in written and oral forms.
You will develop new skills in the numerical solution of partial differential equations, writing numerical algorithms/code and scientific communication. Masters and PhD students will be expected to publish their work in 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 partial differential equations is assumed.
You may be able to apply for a research scholarship in our annual scholarship round.
Contact the Dr Elliot Carr for more information at email@example.com.