For some models instead of attempting to derive an sexy formulation you take random numbers, assign them to certain properties for a given particle and other random numbers to have that particle act. Do this billions of times and you cna build a pretty reliable detailed model of weather patterns or nuclear reactors or whatever.

These supercomputers will rarely be used all at once for a single calculation. Instead the different research groups may be given certain amounts of computation resources according to a set schedule. A big deal at DOE SCs is making sure there isn't idle time. It cost millions to power and cool the systems, and letting them run idle is pretty costly. Same can be said for universities and such.