Study guide: Solving nonlinear ODE and PDE problems

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Combined Picard-Newton algorithm

Idea:

Write a common Picard-Newton algorithm so we can trivially switch between the two methods (e.g., start with Picard, get faster convergence with Newton when $ u $ is closer to the solution)

Algorithm:

Given $ A(u) $, $ b(u) $, and an initial guess $ u^{-} $, iterate until convergence:

solve $ (A + \gamma(A^{\prime}(u^{-})u^{-} + b^{\prime}(u^{-})))\delta u = -A(u^{-})u^{-} + b(u^{-}) $ with respect to $ \delta u $

$ u = u^{-} + \omega\delta u $

$ u^{-}\ \leftarrow\ u $

Note:

$ \gamma =1 $: Newton's method

$ \gamma =0 $: Picard iteration