$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Picard iteration
- Use the most recently computed values \( u^{-} \) of \( u^n \)
in \( \dfc \) and \( f \)
- Or: \( A(u^{-})u=b(u^{-}) \)
- Like solving \( u_t = \nabla\cdot (\dfc(\x)\nabla u) + f(\x,t) \)
Picard iteration in operator notation:
$$ [D_t^- u = D_x\overline{\dfc(u^{-})}^xD_x u
+ D_y\overline{\dfc(u^{-})}^yD_y u + f(u^{-})]_{i,j}^n
$$
