$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Using new notation for implementation
- \( u \) for \( u^n \)
- \( u^{-} \) for \( u^{n,k} \)
- \( u^{(1)} \) for \( u^{n-1} \)
$$ \delta F(\delta u; u^{-}) =- F(u^{-})\quad\hbox{(PDE)}$$
$$
\begin{align*}
F(u^{-}) &= \frac{u^{-} - u^{(1)}}{\Delta t} -
\nabla\cdot (\dfc(u^{-})\nabla u^{-}) + f(u^{-})
\\
\delta F(\delta u; u^{-}) &=
- \frac{1}{\Delta t}\delta u +
\nabla\cdot (\dfc(u^{-})\nabla \delta u) \ + \nonumber\\
&\quad \nabla\cdot (\dfc^{\prime}(u^{-})\delta u\nabla u^{-})
+ f^{\prime}(u^{-})\delta u
\end{align*}
$$
