$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Neumann boundary condition
$$
\begin{equation}
\frac{\partial u}{\partial n} \equiv \normalvec\cdot\nabla u = 0
\tag{21}
\end{equation}
$$
For a 1D domain \( [0,L] \):
$$
\left.\frac{\partial}{\partial n}\right\vert_{x=L} =
\frac{\partial}{\partial x},\quad
\left.\frac{\partial}{\partial n}\right\vert_{x=0} = -
\frac{\partial}{\partial x}
$$
Boundary condition terminology:
