Error after each step in the end of the shooting method for the Bratu problem. I used a methiod of guessing the next initial velocity based on the bisection method for finding a root, increasing the velocity when the curve was below 0 on the right and and decreasing it when it fell below, reducing the amount of change each time it switched. This is by no means the best method for refinement, but it works slowly and surely.

Solution found after 41 attempts to get the error at the right endpoint below 10^-8