Study guide: Solving nonlinear ODE and PDE problems

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Newton's method; Jacobian (1)

Nonlinear eq.no $ i $ has the structure $$ \begin{align*} F_i &= A_{i,i-1}(u_{i-1},u_i)u_{i-1} + A_{i,i}(u_{i-1},u_i,u_{i+1})u_i +\\ &\qquad A_{i,i+1}(u_i, u_{i+1})u_{i+1} - b_i(u_i) \end{align*} $$

Need Jacobian, i.e., need to differentiate $ F(u)=A(u)u - b(u) $ wrt $ u $. Example: $$ \begin{align*} &\frac{\partial}{\partial u_i}(A_{i,i}(u_{i-1},u_i,u_{i+1})u_i) = \frac{\partial A_{i,i}}{\partial u_i}u_i + A_{i,i} \frac{\partial u_i}{\partial u_i}\\ &\quad = \frac{\partial}{\partial u_i}( \frac{1}{2\Delta x^2}(-\dfc(u_{i-1}) + 2\dfc(u_{i}) -\dfc(u_{i+1})) + a)u_i +\\ &\qquad\frac{1}{2\Delta x^2}(-\dfc(u_{i-1}) + 2\dfc(u_{i}) -\dfc(u_{i+1})) + a\\ &\quad =\frac{1}{2\Delta x^2}(2\dfc^\prime (u_i)u_i -\dfc(u_{i-1}) + 2\dfc(u_{i}) -\dfc(u_{i+1})) + a \end{align*} $$