Study guide: Solving nonlinear ODE and PDE problems

$$ \newcommand{\uex}{{u_{\small\mbox{e}}}} \newcommand{\half}{\frac{1}{2}} \newcommand{\tp}{\thinspace .} \newcommand{\Oof}[1]{\mathcal{O}(#1)} \newcommand{\x}{\boldsymbol{x}} \newcommand{\dfc}{\alpha} % diffusion coefficient \newcommand{\Ix}{\mathcal{I}_x} \newcommand{\Iy}{\mathcal{I}_y} \newcommand{\If}{\mathcal{I}_s} % for FEM \newcommand{\Ifd}{{I_d}} % for FEM \newcommand{\basphi}{\varphi} \newcommand{\baspsi}{\psi} \newcommand{\refphi}{\tilde\basphi} \newcommand{\xno}[1]{x_{#1}} \newcommand{\dX}{\, \mathrm{d}X} \newcommand{\dx}{\, \mathrm{d}x} \newcommand{\ds}{\, \mathrm{d}s} $$

Newton's method

$$ F(u)=0,\quad F=(F_S,F_I),\ u=(S,I) $$

Jacobian: $$ \renewcommand*{\arraystretch}{2} J = \left(\begin{array}{cc} \frac{\partial}{\partial S} F_S & \frac{\partial}{\partial I}F_S\\ \frac{\partial}{\partial S} F_I & \frac{\partial}{\partial I}F_I \end{array}\right) = \left(\begin{array}{cc} 1 + \half\Delta t\beta I & \half\Delta t\beta\\ - \half\Delta t\beta S & 1 - \half\Delta t\beta I - \half\Delta t\nu \end{array}\right) $$

Newton system: $ J(u^{-})\delta u = -F(u^{-}) $ $$ \begin{align*} \renewcommand*{\arraystretch}{1.5} & \left(\begin{array}{cc} 1 + \half\Delta t\beta I^{-} & \half\Delta t\beta S^{-}\\ - \half\Delta t\beta S^{-} & 1 - \half\Delta t\beta I^{-} - \half\Delta t\nu \end{array}\right) \left(\begin{array}{c} \delta S\\ \delta I \end{array}\right) =\\ & \qquad\qquad \left(\begin{array}{c} S^{-} - S^{(1)} + \half\Delta t\beta(S^{(1)}I^{(1)} + S^{-}I^{-})\\ I^{-} - I^{(1)} - \half\Delta t\beta(S^{(1)}I^{(1)} + S^{-}I^{-}) - \half\Delta t\nu(I^{(1)} + I^{-}) \end{array}\right) \end{align*} $$