$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Step 4: Formulating a recursive algorithm
- Nature of the algorithm: compute \( u \) in space at \( t=\Delta t, 2\Delta t, 3\Delta t,... \)
- Three time levels are involved in the general discrete equation:
\( n+1 \), \( n \), \( n-1 \)
- \( u^n_i \) and \( u^{n-1}_i \) are then already computed for \( i=0,\ldots,N_x \),
and \( u^{n+1}_i \) is the unknown quantity
Write out \( [D_tD_t u = c^2 D_xD_x]^{n}_i \)
and solve for \( u^{n+1}_i \),
$$
\begin{equation}
u^{n+1}_i = -u^{n-1}_i + 2u^n_i + C^2
\left(u^{n}_{i+1}-2u^{n}_{i} + u^{n}_{i-1}\right)
\tag{9}
\end{equation}
$$
