Finite Difference Methods in Two Dimensions

During my final year at Warwick University I did a project on Numerical Weather Forecasting and one of the methods that came up was the method of finite differences. I recently came across finite differences again over the last few weeks and how they can be used to solve some partial differential equations (PDEs). Here is an example that I found in one of my books that I decided to play around with

Solve, using the method of finite differences, the PDE $$x\dfrac{\partial{f}}{\partial{x}}+(y+1)\dfrac{\partial{f}}{\partial{y}}=0$$ for $0\leq x,y \leq 1$ with $$f(x,0)=x-1$$ $$f(x,1)=\dfrac{x-2}{2}$$ $$f(0,y)=-1$$ $$f(1,y)=-\dfrac{y}{y+1}$$

To solve this problem I used the central difference equations given by

$$\dfrac{\partial{f}}{\partial{x}}\approx \dfrac{f(x+h,y)-f(x-h,y)}{2h}$$

$$\dfrac{\partial{f}}{\partial{y}}\approx \dfrac{f(x,y+k)-f(x,y-k)}{2k}$$

with a step length in both the $x$ and $y$ directions of $\frac{1}{3}$. The domain can be overlaid with a grid as shown in the diagram

This makes the whole problem easier to deal with because now we can see visually what is happening rather than just dealing with everything purely algebraically – there’s no need to make things more difficult than they need to be after all.

The idea is to form a system of four simultaneous equations with $A$, $B$, $C$, and $D$ as the unknown quantities. When the system is solved then the values obtained correspond to the approximate values of $f$ at each of the grid points. Here is my full solution to this problem – PDE Solution Using Finite Differences. This fills in some of the gaps in our knowledge about the function $f$ but there is still a lot of information missing – in particular the points between the grid points.

To resolve this we can make the step-lengths smaller and create more interior grid points. This gives us more information but at a price – for example with a step length of $\frac{1}{5}$ in both directions we would have a system of $16$ simultaneous equations in $16$ unknowns – I don’t really fancy sorting that mess out without the help of a computer. And what’s more – as the step-length gets even smaller we get more and more complicated systems with more unknowns so eventually we would reach a point where even a computer would struggle to do anything with the information.

Nevertheless – finite difference methods are used and are useful for all kinds of problems and can provide a very visual way of numerically solving some otherwise very difficult equations.