Study guide: Solving nonlinear ODE and PDE problems

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Comparison of Newton and Picard iteration

Newton: $$ (A(u^{-}) + A^{\prime}(u^{-})u^{-} + b^{\prime}(u^{-}))\delta u = -A(u^{-})u^{-} + b(u^{-})$$

Rewrite: $$ \underbrace{A(u^{-})(u^{-}+\delta u) - b(u^{-})}_{\hbox{Picard system}} +\, \gamma (A^{\prime}(u^{-})u^{-} + b^{\prime}(u^{-}))\delta u = 0$$

All the "Picard terms" are contained in the Newton formulation.