Description

Stationary points arise in all sorts of practical applications and often a great deal of insight into a system can be gained by finding and/or tracking them as they evolve. As a result there are now a variety of different numerical methods for identifying them with different advantages and disadvantages. Furthemore, the nature of the stationary points (maxima/minima vs saddle points) can require different treatments.

In this project we will investigate some of these methods with a view to applying them to a numerical dataset to gain a deeper understanding of the data's evolution.

We will start with researching different methods for finding stationary points and applying them to test analytical scalar fields. This will involve creating your own python code. Generally, these methods are similar to root finding and take the form of iterative schemes which systematically approach the points. We will then move on to adapting these methods for datasets given on rectangular grids (2D, or even 3D) so as to investigate their behaviour.

(from Wyper & Pontin 2014). The heat map shows thin strips of strong electric current. The X's and O's are numerically identified hyperbolic and elliptic zero's in the magnetic field vector.

An example of such a dataset is shown in the figure, which is a snapshot from a magnetohydrodynamic numerical simulation. The shading shows regions of strong electric current. Here X's and O's correspond to hyperbolic and elliptic zero's in the magnetic field vector. There is a good reason why the X's occur where there is strong current, whereas the O's do not!

Prerequisites

It is strongly advised that you have taken Numerical Analysis II, although it is not essential. The main things needed will be familiarity with iterative methods and plotting. Mathematical Methods in Physics and/or Analysis in Many Variables II could be useful.

Resources

There is plenty of available literature on numerically finding stationary points, see for instance Wikipedia. Some datasets are in the form of vector fields rather than scalars, see for instance this paper by Haynes and Parnell for how to find them in this case.