Differential equations are “trivial” to simulate until they aren’t. Some differential equations feature conserved quantities or invariants that will blow up over long times if you don’t build their conservation into the numerical solver. Some equations can’t be solved with variable time step without careful consideration. Some problems are so huge you can’t solve them without taking some kind of approximation. For instance you might need to find the low dimensional sub space to project onto, and then write the equations of motion for the sub space. You might be solving a finite element problem and need to find the right way to discretize. For electromagnetism this requires writing your elements to be irrotational for the E field.
Furthermore, a lot can be gleaned from understanding the _structure_ of a differential equation. Writing the equation for the RG flow of a field theory and then understanding the fixed points of the flow is crucial. A numerical solution to the RG flow isn’t really that useful.
Basically, once you’ve simulated it—-assuming you’ve done it right—-what actually have you gained? Often less than you want, and generally less insight than you could have gained had you really analyzed it.
What I’m saying is that “trivial to simulate” is only sometimes true. Ansys’s business exists because solving Maxwell’s equations are not “trivial,” and they’re linear!!
Thanks for your detailed reply. While, perhaps not an expert, I am not completely unfamiliar with the difficulties you raise. I have done graduate computational physics and have worked in realtime simulation professionally as well as offline simulation on my own.
"Basically, once you’ve simulated it—-assuming you’ve done it right—-what actually have you gained? "
Usually you now have a computational model that can give answer for any valid input. This may not help your discipline but it certainly helps others. As you suggested with Ansys, there is always room for improvement on these computational models. I would guess unless there was something really wrong with the approach these improvements are going to be algorithmic in nature. Im not saying algorithms dont potentially have some good mathematical analysis behind them but something like say Barnes–Hut feels far more like a computer science data structure than some symbolic mathematical representation.
Furthermore, a lot can be gleaned from understanding the _structure_ of a differential equation. Writing the equation for the RG flow of a field theory and then understanding the fixed points of the flow is crucial. A numerical solution to the RG flow isn’t really that useful.
Basically, once you’ve simulated it—-assuming you’ve done it right—-what actually have you gained? Often less than you want, and generally less insight than you could have gained had you really analyzed it.
What I’m saying is that “trivial to simulate” is only sometimes true. Ansys’s business exists because solving Maxwell’s equations are not “trivial,” and they’re linear!!