NumMethodsPDEs

Numerical Methods for PDEs: Solving PDEs on a computer

=Introduction=

In this tutorial, we'll take a look at how we might model aspects of the world around us on a computer. When we use mathematics to describe many of the phenomena that we see we end up using Partial Differential Equations (PDEs), and so we need ways to solve these numerically. As a notation shorthand, we'll use $$u_x$$ to represent the partial derivative of u with respect to x:


 * $$u_x = \frac{\partial u}{\partial x}$$

=Looking at Nature (& a quick philosophical aside)=

When we observe nature, certain patterns crop up again and again. For example, in the the field of electrostatics, if a function $$f$$ describes a distribution of electric charge, then Poisson's equation:


 * $${\nabla}^2 \varphi = f.$$

gives the electric potential $$\varphi$$.

However, Poisson's equation also describes the steady state temperature of a material when subjected to some heating. Poisson's equation also crops up when considering gravitational potentials. And there's more...

So what's going on here? Is Poisson's equation fundamental in some way? Well, yes I suppose it is. We can look at it another way, however, and say that Poisson's equation is the way we describe steady state phenomena which involve potentials using the language of maths. OK, this sounds a bit out there, but consider the following very general equation for a moment (it's a second-order linear equation in two variables):

$$Au_{xx} + Bu_{xy} + Cu_{yy} + Du_x + Eu_y + Fu = G$$

It turns out that we can categorise certain instances of this equation and then relate these categories to the kind of phenomena that they describe:


 * Parabolic: $$B^2 - 4AC = 0$$.  This family of equations describe heat flow and diffusion processes.
 * Hyperbolic: $$B^2 - 4AC > 0$$.  Describe vibrating systems and wave motion.
 * Elliptic: $$B^2 - 4AC < 0$$.  Steady-state phenomena.

That's pretty handy!

=Discritisation=

Taylor Series:

$$f(x + h) = f(x) + \frac{f'(x)}{1!}h + \frac{f''(x)}{2!}h^2 + \cdots + \frac{f^{(n)}(x)}{n!}h^n + R_n(x)$$