$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Algorithm
- Set initial condition \( u^0_{i,j}=I(x_i,y_j) \)
- Compute \( u^1_{i,j} = \cdots \) for \( i\in\seti{\Ix} \) and \( j\in\seti{\Iy} \)
- Set \( u^1_{i,j}=0 \) for the boundaries \( i=0,N_x \), \( j=0,N_y \)
- For \( n=1,2,\ldots,N_t \):
- Find \( u^{n+1}_{i,j} = \cdots \)
for \( i\in\seti{\Ix} \) and \( j\in\seti{\Iy} \)
- Set \( u^{n+1}_{i,j}=0 \) for the boundaries \( i=0,N_x \), \( j=0,N_y \)
