When I had some algae last spring (granted, it was mustard algae, not black), I used troublefreepool's SLAM method. Basically, I used chlorine to keep the pool's chlorine incredibly high for about 24 hours straight, testing and adjusting every few hours. I threw some yellow out algacide in there at the beginning and a little way through, if I recall correctly. Got rid of that, got the numbers looking good and haven't had a problem since.
Draining seems like an overly expensive solution to the problem. From looking around online, it doesn't seem that black algae should require anything beyond what mustard algae requires, so I'd at least try something like that prior to going through the pain of a refill.