$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Implementation of 2D/3D problems
$$
\begin{align}
u_t &= c^2(u_{xx} + u_{yy}) + f(x,y,t),\quad &(x,y)\in \Omega,\ t\in (0,T]\\
u(x,y,0) &= I(x,y),\quad &(x,y)\in\Omega\\
u_t(x,y,0) &= V(x,y),\quad &(x,y)\in\Omega\\
u &= 0,\quad &(x,y)\in\partial\Omega,\ t\in (0,T]
\end{align}
$$
\( \Omega = [0,L_x]\times [0,L_y] \)
Discretization:
$$ [D_t D_t u = c^2(D_xD_x u + D_yD_y u) + f]^n_{i,j},
$$
