$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Special stencil for the first time step
- The stencil for \( u^1_{i,j} \) (\( n=0 \)) involves \( u^{-1}_{i,j} \)
which is outside the time mesh
- Remedy: use discretized \( u_t(x,0)=V \) and the stencil for \( n=0 \)
to develop a special stencil (as in the 1D case)
$$ [D_{2t}u = V]^0_{i,j}\quad\Rightarrow\quad u^{-1}_{i,j} = u^1_{i,j} - 2\Delta t V_{i,j}
$$
$$ u^{n+1}_{i,j} = u^n_{i,j} -2\Delta V_{i,j} + {\half}
c^2\Delta t^2[D_xD_x u + D_yD_y u]^n_{i,j}$$
