$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Finite difference discretization in a 2D problem
$$ u_t = \nabla\cdot(\dfc(u)\nabla u) + f(u)$$
Backward Euler in time, centered differences in space:
$$ [D_t^- u = D_x\overline{\dfc(u)}^xD_x u
+ D_y\overline{\dfc(u)}^yD_y u + f(u)]_{i,j}^n
$$
$$
\begin{align*}
u^n_{i,j} &- \frac{\Delta t}{h^2}(
\half(\dfc(u_{i,j}^n) + \dfc(u_{i+1,j}^n))(u_{i+1,j}^n-u_{i,j}^n)\\
&\quad -
\half(\dfc(u_{i-1,j}^n) + \dfc(u_{i,j}^n))(u_{i,j}^n-u_{i-1,j}^n) \\
&\quad +
\half(\dfc(u_{i,j}^n) + \dfc(u_{i,j+1}^n))(u_{i,j+1}^n-u_{i,j}^n)\\
&\quad -
\half(\dfc(u_{i,j-1}^n) + \dfc(u_{i,j}^n))(u_{i,j}^n-u_{i-1,j-1}^n))
- \Delta tf(u_{i,j}^n) = u^{n-1}_{i,j}
\end{align*}
$$
Nonlinear algebraic system on the form \( A(u)u=b(u) \)
