$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Picard and Newton iteration in the Crank-Nicolson case
Picard iteration (for a general \( f \)):
$$ \hat F(u) = u - u^{(1)} - \Delta t{\half}f(u^{-},t_{n+1}) -
\Delta t{\half}f(u^{(1)},t_{n})$$
Newton's method:
$$
F(u) = u - u^{(1)} - \Delta t{\half}f(u,t_{n+1}) -
\Delta t{\half}f(u^{(1)},t_{n})
$$
$$ F^{\prime}(u)= 1 - \half\Delta t\frac{\partial f}{\partial u}(u,t_{n+1})$$
