# Computational modelling using exponential integrators

### Study level

Master of Philosophy

Honours

Science and Engineering Faculty

School of Mathematical Sciences

### Topic status

We're looking for students to study this topic.

## Supervisors

Dr Elliot Carr
Position
Senior Lecturer
Division / Faculty
Science and Engineering Faculty
Professor Timothy Moroney
Position
Professor in Mathematics
Division / Faculty
Science and Engineering Faculty

## Overview

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.

### Research activities

You will meet regularly with the supervisory team to acquire new knowledge, brainstorm ideas, discuss your progress and receive feedback and direction on your work. This project will involve pen and paper derivations and calculations, developing code in MATLAB and communicating your work in written form.

### Outcomes

You will develop new skills in:
• numerical linear algebra
• time discretisation methods
• partial differential equations
• programming (MATLAB)
• mathematical writing
• LaTeX typesetting.
New techniques/knowledge/results generated during your project will be published in leading mathematical/scientific journals.

### Skills and experience

This project can be tailored to your study level whether you are an Honours or PhD student. This topic can also be personalised to suit your individual interests and skills. Ideally, the successful student will have some prior experience with solving ordinary or partial differential equations in MATLAB.

### Scholarships

You may be able to apply for a research scholarship in our annual scholarship round.

Annual scholarship round