$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Combined Picard and Newton formulation
$$
\begin{align*}
& \frac{u - u^{(1)}}{\Delta t} =
\nabla\cdot (\dfc(u^{-})\nabla u) + f(u^{-}) + \\
&\qquad \gamma(\nabla\cdot (\dfc^{\prime}(u^{-})(u - u^{-})\nabla u^{-})
+ f^{\prime}(u^{-})(u - u^{-}))
\end{align*}
$$
Observe:
- \( \gamma=0 \): Picard iteration
- \( \gamma=1 \): Newton's method
Why is this formulation convenient?
Easy to switch (start with Picard, use Newton close to solution)
