Many mathematical models that describe problems in science and engineering are defined in high dimensional spaces. Examples can be found in a diverse range of application areas such as quantum chemistry, kinetic theory descriptions of materials (including complex fluids), the chemical master equation governing many biological processes (e.g. cell signalling) and models of financial mathematics (e.g. option pricing). The mathematical description of these problems is invariably in terms of a system of partial differential equations (PDEs). For practical problems these systems do not possess analytical solutions and therefore it is necessary to solve them numerically. It is important that the system of algebraic equations obtained as a result of discretisation is a compatible (mimetic) and physically consistent system so that the numerical approximation is an accurate representation of the physical solution to the problem.
The governing systems of PDEs is written in terms of an equivalent system of first-order differential equations which is subsequently formulated in terms of a least squares functional. Effectively the solution of a system of PDEs is converted into an unconstrained minimisation problem.
The exceptional stability of least-squares formulations has led to the widespread use of low-order finite elements in their discretization. Unfortunately, these methods are only approximately conservative, which generally leads to violation of fundamental physical properties, such as loss of mass conservation. In many cases this drawback can outweigh the potential advantages of least squares methods. As a result, improving the conservation properties of least-squares methods is crucially important.
The School of Mathematics provides an outstanding postgraduate research environment. The latest PRES ranks its overall satisfaction 3rd and its professional development 1st (out of 22). While project-specific academic training will be provided by the supervisors, the student will also benefit from the School’s seminar culture, access to three national course centres in mathematics, statistics and OR (MAGIC, APTS, NATCOR) as well as Cardiff’s Doctoral Academy, which offers a comprehensive programme for postgraduate researchers to develop their professional skills. In addition, the SIAM-IMA Student Chapter is a valuable forum for exchange of ideas and public engagement. Our students are encouraged to contribute to engagement activities such as STEM Live!; “Mathemagic” a 12 day event run in collaboration with the Operational Research Society and Techniquest and Soapbox Science. A monthly Women in Maths forum has also been established by a group of PhD students which is open to students at all levels.
The 3.5 year studentship includes UK/EU fees, stipend (amount for 2020/21 is £15285) and a research training grant to cover costs such as research consumables, training, conferences and travel.
The studentship represents an excellent opportunity to conduct innovative research in the fields of numerical analysis, continuum mechanics and computational fluid dynamics. The research will lead to original contributions to the theoretical and computational development of efficient and accurate numerical methods. This will place the student in an excellent position to progress their academic career on completion of the project. HOW TO APPLY
Applicants should apply through the Cardiff University online application portal
, for a Doctor of Philosophy in Mathematics with an entry point of October 2020
In the research proposal section of your application, please specify the project title and supervisors of this project. In the funding section, please select "I will be applying for a scholarship / grant" and specify that you are applying for advertised funding from EPRSC DTP.