$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Newton's method: the nonlinear algebraic equations
Define the nonlinear equations (use \( u \) for \( u^n \), \( u^{(1)} \) for \( u^{n-1} \)):
$$
\begin{align*}
F_{i,j} &= u_{i,j} - \frac{\Delta t}{h^2}(\\
&\qquad \half(\dfc(u_{i,j}) + \dfc(u_{i+1,j}))(u_{i+1,j}-u_{i,j}) -\\
&\qquad \half(\dfc(u_{i-1,j}) + \dfc(u_{i,j}))(u_{i,j}-u_{i-1,j}) + \\
&\qquad \half(\dfc(u_{i,j}) + \dfc(u_{i,j+1}))(u_{i,j+1}-u_{i,j}) -\\
&\qquad \half(\dfc(u_{i,j-1}) + \dfc(u_{i,j}))(u_{i,j}-u_{i-1,j-1})) -
\Delta t\, f(u_{i,j}) - u^{(1)}_{i,j} = 0
\end{align*}
$$
