I am working with chapter 2 in LeVeque's book: https://faculty.washington.edu/rjl/fdmbook/
I have build my own solver in Python to solve the 2 point BVP: $$ \epsilon u''+u(u'-1) =0 , \\ u(0)=\alpha, u(1)= \beta $$
I have followed the axact steps as described by the answer by VoB in this post: Non-linear Boundary Value Problem. How to compute the Jacobian? (uniform grid; Newton method solve for $G(U)=0$ wrt U).
For the parameters $[\epsilon, \alpha, \beta ] = [0.01, -1,1.5]$ I get this result
In the 10th run it has converged to what I believe is the true solution. This looks pretty much correct to me. (run1 is the intial guess in Newton steps). My stepsize is $\Delta x = h = 0.005$
In the post it is mentioned that the order of convergence is 2. But how do I compute the errors to determine the order of convergence to be 2?