$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Neumann condition and a variable coefficient
Consider \( \partial u/\partial x=0 \) at \( x=L=N_x\Delta x \):
$$ \frac{u_{i+1}^{n} - u_{i-1}^n}{2\Delta x} = 0\quad u_{i+1}^n = u_{i-1}^n,
\quad i=N_x
$$
Insert \( u_{i+1}^n=u_{i-1}^n \) in the stencil
(30)
for \( i=N_x \) and obtain
$$
u^{n+1}_i \approx
- u_i^{n-1} + 2u_i^n + \left(\frac{\Delta x}{\Delta t}\right)^2
2q_{i}(u_{i-1}^n - u_{i}^n) + \Delta t^2 f^n_i
$$
(We have used \( q_{i+\half} + q_{i-\half}\approx 2q_i \).)
Alternative: assume \( dq/dx=0 \) (simpler).
