$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Implicit Crank-Nicolson discretization
Crank-Nicolson discretization:
$$ [D_t u = u(1-u)]^{n+\half}$$
$$
\frac{u^{n+1}-u^n}{\Delta t} = u^{n+\half} -
(u^{n+\half})^2
$$
Approximate \( u^{n+\half} \) as usual by an arithmetic
mean,
$$ u^{n+\half}\approx \half(u^n + u^{n+1})$$
$$ (u^{n+\half})^2\approx \frac{1}{4}(u^n + u^{n+1})^2\quad\hbox{(nonlinear term)}$$
which is nonlinear in the unknown \( u^{n+1} \).
