Study guide: Solving nonlinear ODE and PDE problems

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Newton's method for Backward Euler scheme

Newton's method requires the computation of the derivative $$ F^{\prime}(u) = 1 - \Delta t\frac{\partial f}{\partial u}(u,t_n)$$

Algorithm for Newton's method for $ u^{\prime}=f(u,t) $.

Start with $ u^{-}=u^{(1)} $, then iterate $$ u = u^{-} - \omega \frac{F(u^{-})}{F^{\prime}(u^{-})} = u^{-} - \omega \frac{u^{(1)} + \Delta t\, f(u^{-},t_{n})}{1 - \Delta t \frac{\partial}{\partial u}f(u^{-},t_n)} $$