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 [math]\displaystyle{ u_x }[/math] to represent the partial derivative of u with respect to x:

[math]\displaystyle{ u_x = \frac{\partial u}{\partial x} }[/math]

Looking at Nature

When we observe nature, certain patterns crop up again and again. For example, in the the field of electrostatics, if a function [math]\displaystyle{ f }[/math] describes a distribution of electric charge, then Poisson's equation:

[math]\displaystyle{ {\nabla}^2 \varphi = f. }[/math]

gives the electric potential [math]\displaystyle{ \varphi }[/math].

A second-order linear equation in two variables:

[math]\displaystyle{ Au_{xx} + Bu_{xy} + Cu_{yy} + Du_x + Eu_y + Fu = G }[/math]

• Parabolic: [math]\displaystyle{ B^2 - 4AC = 0 }[/math]. This family of equations describe heat flow and diffusion processes.
• Hyperbolic: [math]\displaystyle{ B^2 - 4AC \gt 0 }[/math]. Describe vibrating systems and wave motion.
• Elliptic: [math]\displaystyle{ B^2 - 4AC \lt 0 }[/math]. Steady-state phenomena.

Taylor Series:

[math]\displaystyle{ 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) }[/math]