# Finite difference methods

1. Nov 30, 2005

### Zurtex

We have been given a program that can solve the following equation using finite difference methods:

$$- \epsilon \left( \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} \right) - \frac{\partial \phi}{\partial x} = \sin \left(\pi y\right) \quad x,y \, \in \, (0,1)$$

This is a convection diffusion model of ocean currents where the x-direction is east and the y-direction is north. And $\phi = 0$ on the boundaries.

I’ve managed to do most the problems for this but we asked to investigate the case when the Earth is rotating in the opposite direction, by changing the sign of the convection term. However I am unsure what that term is. Also I’m a little dodgy on my finite difference methods for partial differential equations, how exactly do they differ from working out finite difference methods for ordinary differential equations?

Any help at all will be greatly appreciated.

2. Nov 30, 2005

### Zurtex

Reading over my notes and thinking about it, I think I more or less have the hang of getting finite differences for PDEs, but could someone tell me which is the convection term please.

3. Dec 2, 2005

### saltydog

Suppose you got this already Zurtex. I think about it's relevance perhaps to the north atlantic conveyor belt. You know that story? Well, it starts off about a man name Jed, a poor mountianeer where he kept . . . wait, wrong story. It's that river of ocean water that circles the globe, takes about 20,000 years for one circuit. Sometimes it can be disrupted by a large influx of fresh water, glaicers melting, ice wall breaking, causing large changes in climate. Wouldn't it be interesting if a non-linear PDE could model this with a critical point expressing this abrupt disruption. Anyway, I think it's the

$$\frac{\partial \phi}{\partial x}$$

term.

Oh yea, I think Mathematica could do a nice job of running a finite difference say with 10,000 equations (i.e. increment of 0.01).