Study guide: Solving nonlinear ODE and PDE problems

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Implicit time discretization

A Crank-Nicolson scheme: $$ \begin{align*} \frac{S^{n+1}-S^n}{\Delta t} &= -\beta [SI]^{n+\half} \approx -\frac{\beta}{2}(S^nI^n + S^{n+1}I^{n+1})\\ \frac{I^{n+1}-I^n}{\Delta t} &= \beta [SI]^{n+\half} - \nu I^{n+\half} \approx \frac{\beta}{2}(S^nI^n + S^{n+1}I^{n+1}) - \frac{\nu}{2}(I^n + I^{n+1}) \end{align*} $$

New notation: $ S $ for $ S^{n+1} $, $ S^{(1)} $ for $ S^n $, $ I $ for $ I^{n+1} $, $ I^{(1)} $ for $ I^n $ $$ \begin{align*} F_S(S,I) &= S - S^{(1)} + \half\Delta t\beta(S^{(1)}I^{(1)} + SI) = 0\\ F_I(S,I) &= I - I^{(1)} - \half\Delta t\beta(S^{(1)}I^{(1)} + SI) - \half\Delta t\nu(I^{(1)} + I) =0 \end{align*} $$