Study guide: Finite difference methods for wave motion

$$ \newcommand{\uex}{{u_{\small\mbox{e}}}} \newcommand{\half}{\frac{1}{2}} \newcommand{\halfi}{{1/2}} \newcommand{\xpoint}{\boldsymbol{x}} \newcommand{\normalvec}{\boldsymbol{n}} \newcommand{\Oof}[1]{\mathcal{O}(#1)} \newcommand{\Ix}{\mathcal{I}_x} \newcommand{\Iy}{\mathcal{I}_y} \newcommand{\It}{\mathcal{I}_t} \newcommand{\setb}[1]{#1^0} % set begin \newcommand{\sete}[1]{#1^{-1}} % set end \newcommand{\setl}[1]{#1^-} \newcommand{\setr}[1]{#1^+} \newcommand{\seti}[1]{#1^i} \newcommand{\Real}{\mathbb{R}} $$

The Courant number

$$ \begin{equation} C = c\frac{\Delta t}{\Delta x}, \end{equation} $$ is known as the (dimensionless) Courant number

Observe.

There is only one parameter, $ C $, in the discrete model: $ C $ lumps mesh parameters $ \Delta t $ and $ \Delta x $ with the only physical parameter, the wave velocity $ c $. The value $ C $ and the smoothness of $ I(x) $ govern the quality of the numerical solution.