I have a second order, non-linear ODE where the coefficients jump at a point depending on the solution itself. I have tried picard-type iteration with a simple upwinding scheme and dynamic mesh refinement, but both of these introduce their own errors specific to the problem.
A simple example looks like
$ a(x, y)y'' + b(x, y)y' + c(x, y)y = 0 $
where the coefficients vary smoothly except at some point $\alpha(x, y)$ which is part of the solution, $y$ is continuous, and where boundary values are given.
Does anyone know of references or have any ideas for how to solve this type of problem?