$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Scalar computations: arrays
Store \( u^{n+1}_{i,j} \), \( u^{n}_{i,j} \), and
\( u^{n-1}_{i,j} \) in three two-dimensional arrays:
u = zeros((Nx+1,Ny+1)) # solution array
u_1 = zeros((Nx+1,Ny+1)) # solution at t-dt
u_2 = zeros((Nx+1,Ny+1)) # solution at t-2*dt
\( u^{n+1}_{i,j} \) corresponds to u[i,j], etc.
