I always heard soaking in hot water and baking soda would do the trick.
Fill half the pot with water, add 3-4 tablespoons of baking soda and bring to a boil, turn off the heat, let it soak for 30 minutes. If after that you still have some black spots, use Bar Keepers Friend (for stainless steel pots), I love that stuff!
Sincerely hope this helps.
"We are what we repeatedly do. Excellence, then, it is not an act, but a habit." Aristotle