$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Step 3: Algebraic version of the initial conditions
- Need to replace the derivative in the initial condition
\( u_t(x,0)=0 \) by a finite difference approximation
- The differences for \( u_{tt} \) and \( u_{xx} \) have second-order accuracy
- Use a centered difference for \( u_t(x,0) \)
$$ [D_{2t} u]^n_i = 0,\quad n=0\quad\Rightarrow\quad
u^{n-1}_i=u^{n+1}_i,\quad i=0,\ldots,N_x$$
The other initial condition \( u(x,0)=I(x) \) can be computed by
$$ u_i^0 = I(x_i),\quad i=0,\ldots,N_x$$
