$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Manual linearization for a given \( f(u,t) \)
- \( f(u^{-},t) \): explicit treatment of \( f \)
(as in time-discretization)
- \( f(u,t) \): fully implicit treatment of \( f \)
- If \( f \) has some structure, say \( f(u,t)=u^3 \), we may
think of a partially implicit treatment: \( (u^{-})^2u \)
- More implicit treatment of \( f \) often gives faster
convergence
(as it gives more stable time discretizations)
Trick for partially implicit treatment of a general \( f(u,t) \):
$$ f(u^{-},t)\frac{u}{u^{-1}} $$
(Idea: \( u\approx u^{-} \))