If you want to see a good example of one of the most challenging scientific problems in modern science, check out the TED talk by Gavin Schmidt, who is modeling the earth's climate. See what goes into a computer simulation in terms of the math, and then how the simulation can be tested for validity and accuracy based on real data and measurements from past climate trends. Finally, and this is the power of a simulation that has been tested and has a high level of confidence it is doing what we think it is doing, he shows how the simulation can make predictions for future climate trends depending on what values the parameters in the simulation are given.
Good computer simulations allow scientists to do computer experiments on phenomena and systems they otherwise could not experiment on. Imagine trying to figure anything out about the way a star works - we obviously cannot directly test a star, or star systems. But astrophysicists can run computer simulations using the known laws of physics and compare the results to what is really out there. If there is a good match with the simulation, then the simulations can be used to see what would happen under different conditions, and predict other objects and systems that may not have even been observed yet!
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