$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Picard iteration with alternative notation
$$
\frac{u^{n,k+1} - u^{n-1}}{\Delta t} = \nabla\cdot (\dfc(u^{n,k})
\nabla u^{n,k+1})
+ f(u^{n,k})
$$
Rewrite with a simplified, implementation-friendly notation:
- \( u \) means the unknown \( u^{n,k+1} \) to solve for
- \( u^{-} \) means the most recent approximation to \( u \)
- \( u^{(1)} \) means \( u^{n-1} \) (\( u^{(\ell)} \) means \( u^{n-\ell} \))
$$
\frac{u - u^{(1)}}{\Delta t} = \nabla\cdot (\dfc(u^{-})
\nabla u)
+ f(u^{-})
$$
Start iteration with \( u^{-}=u^{(1)} \);
update with \( u^{-} \) to \( u \).
