We tried using the ADI (1st ed.) and SOR routines in ch. 19, and noticed the routines do not solve for the solution on the boundaries. Is this correct or are we missing something? We have not been able to get either routine to work.

The routines are for boundary conditions where the solution is zero on the boundary. If you have different boundary conditions, you must first rewrite the problem to have zero boundary conditions following the description in the text.

Saul Teukolsky