$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Linearization by a geometric mean
Using a geometric mean for \( (u^{n+\half})^2 \) linearizes
the nonlinear term \( (u^{n+\half})^2 \) (error \( \Oof{\Delta t^2} \) as
in the discretization of \( u^{\prime} \)):
$$ (u^{n+\half})^2\approx u^nu^{n+1}$$
Arithmetic mean on the linear \( u^{n+\frac{1}{2}} \) term and a geometric
mean for \( (u^{n+\half})^2 \) gives a linear equation
for \( u^{n+1} \):
$$ \frac{{\color{red}u^{n+1}}-u^n}{\Delta t} =
\half(u^n + {\color{red}u^{n+1}}) + u^n{\color{red}u^{n+1}}$$
Note: Here we turned a nonlinear algebraic equation into a linear
one. No need for iteration! (Consistent \( \Oof{\Delta t^2} \) approx.)
