MATH4PD Numerical Solution of PDEs
Not offered in 2016
While analytic solutions to partial differential equations (PDEs) — e.g. Fourier series, integral transforms, Green’s functions — coupled with qualitative analysis — e.g. maximum principles, asymptotic behaviour — provide essential insights into mathematical models, most realistic models require numerical solutions. Applications are numerous and range from modelling astrophysical detonations (shocks), elastic deformations (e.g. seismic waves) and fluid flow (e.g. subsurface flow–groundwater, geothermal) to medical and geophysical imaging.
The aim of the course is to give a broad introduction to numerical methods for solving PDEs, and prepare you for graduate studies in applied mathematics. We will cover the major methods. By the end of the course you should be able to code your own schemes for 1 and 2-D problems, and be able to analyse convergence and stability. A first course on PDEs is essential, and some exposure to Matlab and linear algebra will be required. Background in analysis would be advantageous, will help you appreciate the core ideas behind numerical techniques, and put you in a better position to carry out independent work.
We aim to cover the following topics:
- Finite difference schemes for parabolic and hyperbolic systems
- Finite element method for elliptic equations
- Iterative techniques for solving systems of linear equations (including multi-grid)
We may, if time permits, explore some of the following topics:
- Finite element method for parabolic equations
- Finite volume schemes
- Spectral methods
- Free boundary value problems
Learning will be hands on, and homework will consist, in the main, of Matlab based mini-projects.
Linear Algebra (MATH 202) and Partial Differential Equations (MATH 304) or equivalent
Nick Dudley Ward (Physics department, phone 479 7808, email: email@example.com)
(Nick is director of the Otago Computational Modelling Group (OCMO), www.ocmo.co.nz)
There will be assignments and a project. Details to come...
Details to come...