$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Variable coefficients (1)
3D wave equation:
$$ \varrho u_{tt} = (qu_x)_x + (qu_y)_y + (qu_z)_z + f(x,y,z,t) $$
Just apply the 1D discretization for each term:
$$
\begin{equation}
[\varrho D_tD_t u = (D_x\overline{q}^x D_x u +
D_y\overline{q}^y D_yu + D_z\overline{q}^z D_z u) + f]^n_{i,j,k}
\end{equation}
$$
Need special formula for \( u^1_{i,j,k} \)
(use \( [D_{2t}u=V]^0 \) and stencil for \( n=0 \)).
