The starting point of the mathematical theory of solitons for the Korteweg-de Vries equation was the numerical experiment of Kruskal and Zabusky in 1965, showing that solitons of different amplitudes, hence traveling at different speeds, crossed each other and reemerged (almost) undisturbed. I think this is an appropriate example in this thread, since this is an actual new phenomenon, totally unespected, discovered by computer simulation, then rigorously proved and widely generalized to constitute a whole new mathematical theory.