To do something convulsively is to do it in an irregular, jerky way. If you can't help laughing convulsively during a friend's awkward speech, you'll be shaking with laughter.
Movements that are connected to actual convulsions — the involuntary, abnormal movement of the body's muscles — happen convulsively. Other things occur convulsively too, like hiccups and laughter and crying: "She sobbed convulsively when her dog ran away." The adverb convulsively comes from the verb convulse, which is rooted in the Latin convulsus, "pulled away" or "wrenched."