Trapezoidal method using fixed point iteration

Question

I am not sure how to apply the trapezoidal method using fixed point iteration, at each step, to this equation $\frac{dy}{dt}=\cos{(\frac{2y}{4})}$.

Any help will be appreciated as I've been stuck on this for the past few days.

Answer 1

The implicit trapezoidal method for $y'=f(y)$ reads as $$ \frac{y_{n+1}-y_n}{h}=\frac{f(y_{n+1})+f(y_n)}2. $$ The usual fixed-point equation formulation of that is $$ y_{n+1}=y_n+\frac{h}2(f(y_{n+1})+f(y_n))=r_n+\frac{h}2f(y_{n+1}) $$ That is, you compute $r_n=y_n+\frac{h}2f(y_n)$, an then iterate $$ z^1=y_n+hf(y_n),~~~ z^{j+1}=r_n+\frac{h}2f(z^j), $$ and set $y_{n+1}=z^j$ after the changes in the iteration become small enough.

This converges if $Lh<2$ where $L$ is a Lipschitz constant of $f$.