Loading [MathJax]/extensions/TeX/boldsymbol.js
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
Computing the first step
- The formula breaks down for u^1 because u^{-1} is unknown and
outside the mesh!
- And: we have not used the initial condition u^{\prime}(0)=0 .
Discretize u^{\prime}(0)=0 by a centered difference
\frac{u^1-u^{-1}}{2\Delta t} = 0\quad\Rightarrow\quad u^{-1} = u^1
Inserted in the scheme for n=0 gives
u^1 = u^0 - \half \Delta t^2 \omega^2 u^0
