We often end up sitting in 正座 (seiza) for some time during class. This is much longer during demonstrations and gradings. In addition, a whole corpus of techniques (suwari waza, ghosin no kata) do start from seiza.
Most of us westerners did not have the benefit of sitting in seiza during our childhood, thus prolonged periods of time are just painful. In my case, it's the legs that go to sleep and get really stiff.
Is there anything one can do to increase one's ability to cope with sitting in seiza?
To get better at something you have to do more of that thing. This is kind of obvious. It is the how and why a specific set of exercises will improve safely my ability to sit in seiza that I am interested in.
⚠ None of the current answers go beyond "just do more of it" which is lazy advise. Clearly, the way to get better at something is to do more of that. That is obvious. I am looking for an effective and safe way to improve. The bounty was not awarded to anyone.