$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Neumann boundary condition in 2D
Use ideas from 1D! Example: \( \frac{\partial u}{\partial n} \) at \( y=0 \),
\( \frac{\partial u}{\partial n} = -\frac{\partial u}{\partial y} \)
Boundary condition discretization:
$$ [-D_{2y} u = 0]^n_{i,0}\quad\Rightarrow\quad \frac{u^n_{i,1}-u^n_{i,-1}}{2\Delta y} = 0,\ i\in\Ix
$$
Insert \( u^n_{i,-1}=u^n_{i,1} \) in the stencil for \( u^{n+1}_{i,j=0} \) to
obtain a modified stencil on the boundary.
Pattern: use interior stencil also on the bundary, but replace
\( j-1 \) by \( j+1 \)
Alternative: use ghost cells and ghost values
