$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
A Picard iteration
- We have approximations \( S^{-} \) and \( I^{-} \) to \( S \) and \( I \).
- Linearize \( SI \) in \( S \) ODE as \( I^{-}S \) (linear equation in \( S \)!)
- Linearize \( SI \) in \( I \) ODE as \( S^{-}I \) (linear equation in \( I \)!)
$$
\begin{align*}
S &= \frac{S^{(1)} - \half\Delta t\beta S^{(1)}I^{(1)}}
{1 + \half\Delta t\beta I^{-}}
\\
I &= \frac{I^{(1)} + \half\Delta t\beta S^{(1)}I^{(1)}}
{1 - \half\Delta t\beta S^{-} + \nu}
\end{align*}
$$
Before a new iteration: \( S^{-}\ \leftarrow\ S \) and
\( I^{-}\ \leftarrow\ I \)
