Strong consistency of finite difference approximations to PDEs

Director of Studies Dr Daniel Robertz (daniel.robertz@plymouth.ac.uk, tel.: +44-1752-586334)

2nd Supervisor Dr Jason Hughes (j.hughes@plymouth.ac.uk, tel.: +44-1752-586887)

Applications are invited for a three-year MPhil/PhD studentship. The studentship will start on 1 October 2021.

Project description 

The methods of numerical analysis for solving differential equations are typically based on approximations of these equations by difference equations. The processes of generating finite difference schemes and of estimating their quality should be simplified and automatized by methods of computer algebra. 

A first step in the direction of assessing, with the aid of symbolic computation, a difference approximation in the linear case was accomplished by V. P. Gerdt and D. Robertz by using Gröbner bases for both differential and difference equations. For the novel notion of strong consistency it is to be verified that the limit of the difference Gröbner basis is a consequence of the linear PDE system (and thus, of the differential Gröbner basis) when the mesh steps of the approximation tend to zero. 

The aims of the proposed project are:

• To develop an algorithmic approach based on difference and differential algebra for the generation of strongly consistent finite difference schemes for systems of nonlinear PDEs.
• To develop the first symbolic software package for computing decompositions of nonlinear difference systems. This will be a package analogous to one that has already been developed for systems of nonlinear PDEs, but it requires new algorithmic ideas to render computations in the difference case viable. The new package will facilitate the automatic check of strong consistency of difference schemes.
• To apply this approach to systems of PDEs of interest, from fluid dynamics in particular, and thereby demonstrate the computational benefits of the new concept.

Eligibility 

Applicants should have (at least) a first or upper second class honours degree in an appropriate subject and preferably a relevant MSc or MRes qualification. 

The studentship is supported for three years and includes full home tuition fees plus a stipend of £15,609 per annum (2021/22 rate). The studentship will only fully fund those applicants who are eligible for home fees with relevant qualifications. Applicants normally required to cover international fees will have to cover the difference between the home and the international tuition fee rates (approximately £12,697 per annum).

If you wish to discuss this project further informally, please contact Dr Daniel Robertz, daniel.robertz@plymouth.ac.uk. However, applications must be made in accordance with the details shown below.

General information about applying for a research degree at the University of Plymouth and to apply for this position please visit: https://www.plymouth.ac.uk/student-life/your-studies/research-degrees. 

Please mark it FAO Doctoral College and clearly state that you are applying for a PhD studentship within the School of Engineering, Computing and Mathematics.

For more information on the admissions process email the Doctoral College, doctoralcollege@plymouth.ac.uk.

The closing date for applications is 19 April 2021. Shortlisted candidates will be invited for interview the week beginning 3 May 2021. We regret that we may not be able to respond to all applications. Applicants who have not received an offer of a place by 31 May should consider their application has been unsuccessful on this occasion.