Part of the Texts in Applied Mathematics book series (TAM, volume 52)

## Abstract

We now begin the study of numerical wave propagation. Everyone has experience with traveling waves, whether it is waves on a lake, sound waves, or perhaps an earthquake or two. It is not particularly difficult to write down a reasonable-looking finite difference approximation to a wave equation. Most of the effort is invested in trying to determine whether the method actually works. There are unique complications for numerical wave propagation, and so to introduce the ideas we use one of the simplest mathematical equations that produces traveling waves. This is the advection equation, given as
$$\frac{{\partial u}} {{\partial t}} + a\frac{{\partial u}} {{\partial x}} = 0,{\text{ }}for\left\{ \begin{gathered} - \infty < x < \infty , \hfill \\ 0 < t, \hfill \\ \end{gathered} \right.$$
(4.1)
where u(x, 0) = g(x). It is assumed that a is a positive constant.

## Keywords

Heat Equation Truncation Error Method Stable Upwind Scheme Implicit Method