Many physical processes can be modelled using time and space dependent partial differential equations. Solutions to these equations provide 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 nonlinearities in the governing equations and the irregular geometries on which they apply, exact analytical solutions are almost always not accessible. In such cases, numerical 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.
- numerical linear algebra
- time discretisation methods
- partial differential equations
- programming (MATLAB)
- mathematical writing
- LaTeX typesetting.
Skills and experience
You may be able to apply for a research scholarship in our annual scholarship round.
Contact the supervisor for more information.