Nature of project: **software**, theory

Available to students on full-time physics degree schemes or joint students.

The heat equation, also referred to as the heat diffusion equation or diffusion equation, is a single partial differential equation with features that mirror or underpin numerous elements in other, more complex coupled systems like the Navier-Stokes Equations and Magnetohydrodynamic Equations. Comparing numerical schemes as they are implemented to solve the heat equation is, therefore, a effective method by which to understand those numerical schemes.

This project proposes to analyse the efficacy of Smoothed Particle (SP) algorithms against the traditional finite difference method(s). The practical component of the project focuses on the implementation of a smoothed particle solution to the heat equation in 1D and analysing the behaviour of the solution as it compares to solutions generated by other, provided algorithms.

*A successful project will develop beyond the above in one/some of the following directions:*

The project can, naturally, extend into a multidimensional solution to the heat equation (in 2D and 3D). However, in addition, the project can be expanding by implementing variations on the base smoothed particle method using variable spacial basis, alternative approximations of second-order derivatives and alternative boundary conditions.

When considering where to take your project, please bear in mind the time available. It is preferable to do fewer things well than to try many and not get conclusive results on any of them. However, sometimes it is useful to have a couple of strands of investigation in parallel to work on in case delays occur.

*This project is only available as a Y3 project.*

Please speak to **Tom Knight** if you consider doing this project.

*Initial literature for students:*

- J. Monaghan, 2005: Smoothed Particle Hydrodynamics. Reports on Progress in Physics, 68, 1703.
- D. Kincaid & W. Cheney, 1990: Numerical Analysis: Mathematics of Scientific Computing (*third edition).
- R. Gingold & J. Monaghan, 1977: Smoothed Particle hydrodynamics - Theory and Application to Non-Spherical Stars. Monthly Notices of the Royal Astronomical Society, 181, 376-389.
- T. Knight, 2013: Smoothed Particle Magnetohydrodynamics for the Solar Corona, Aberystwyth University [thesis].

Programming heavy project. Previous python (or other object orientated language) experience would be useful though not necessary.

Difficulty is dependent on the student's pace through the project and the chosen extensions.

milestone | to be completed by |
---|---|

Understanding of the smoothed particle method and derived the smoothed particle heat equation. | end of November |

.Working smoothed particle code | end of February |

Generated solutions to the heat equation using smoothed particle and finite difference codes. | mid-March |

Comparison completed (and possible extensions implemented). | Easter |