Last updated: 24/03/2000

I am busy working on a numerical project for my linear and non-linear waves course. At the moment I'm modelling two non-linear pde's by an explicit finite elements method. Basically I'm looking at the Non-linear Schroedinger (NLS) equation which is i*dU/dt + d^2 U/dx^2 + |U|^2U = 0 and the Sine-gordon equation which is d^2 U/dt^2 - d^2 U/dx^2 + sin(u) =0 The simplest method of modelling these partial differential equations on a computer is by looking at the corresponding partial difference equations which result from the fact that on a computer we can only approximate derivatives by Newton quotients for some finite dt or dx, i.e. we cannot take the limit of dt -> 0 or dx -> 0. This method is the simplest because it is the natural way one would think of implementing these equations. However this approach not optimal at all since for good results one needs really small dx's which means long computing times. Also for the method to be stable at all there are limits on the size of dt which is usually bounded above by dx^2 or dx^3 depending on the order of your PDE, which means even longer running times. There are many more sophisticated methods using spectral techniques etc... but at my level the above analysis was sufficient. Anyway, that's the method, let me now show you the pictures.

This is a "breather" of the sine-gordon equation.

I wanted to make more pictures but for some reason Photoshop kept copying only part of the palette from matlab so half the colours were missing on the pictures which made them look crap. I couldn't figure out how to fix it and wasted too much time on it already anyway, so I have to leave it like this.

Dune News!

5 May 2000