Loading [MathJax]/extensions/TeX/boldsymbol.js
\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}}
Step 2: Fulfilling the equation at the mesh points
Let the PDE be satisfied at all interior mesh points:
\begin{equation}
\frac{\partial^2}{\partial t^2} u(x_i, t_n) =
c^2\frac{\partial^2}{\partial x^2} u(x_i, t_n),
\tag{6}
\end{equation}
for i=1,\ldots,N_x-1 and n=1,\ldots,N_t-1 .
For n=0 we have the initial conditions u=I(x) and u_t=0 ,
and at the boundaries i=0,N_x we have the boundary condition u=0 .
