$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Alternative implementation via ghost cells
- Instead of modifying the stencil at the boundary,
we extend the mesh to cover \( u_{-1}^n \) and \( u_{N_x+1}^n \)
- The extra left and right cell are called ghost cells
- The extra points are called ghost points
- The \( u_{-1}^n \) and \( u_{N_x+1}^n \) values are called ghost values
- Update ghost values as \( u_{i-1}^n = u_{i+1}^n \) for \( i=0 \) and \( i=N_x \)
- Then the stencil becomes right at the boundary
