So in my setting there is a small and exclusive group of people that have access to devices that work like computers (but which are not the same as computers. They do not use semiconductors and electricity, for instance). These people would use these machines to simulate the world and acquire knowledge of physics and science, putting them way ahead of their time. These machines are condemmed by organized religion because they are borderline prophecy and magic for those seeing it from outside, so these people go about their business in secret.
This particular story tells the tale of Julius, a lad who loses his parents for a plague and is sent to live with his grandfather's brother, a bitter old man with a lot of money named Cesare who seem to always be on guard. The boy arrives at the countryside home of Cesare and is treated like an irritating guest.
The story follows the emotional bonding of Cesare and Julius as he comes as a guest and then slowly progresses towards family. Cesare, the bitter man finaly melts his protections and see in Julius the son he never had and Julius finds in Cesare the guardian that he lost. The entire piece resounds with themes of family union and filial love but also growing up as Julius goes from a naive lad to an able scientific apprencite and lab assistent and seem posed to carry on the work of Cesare, the son the man never had.
Although the themes are important, the backdrop on which they are dramatized is one of science and discovery. The emotional journey of Cesare and Julius happens as Julius goes down into Cesare's research. He finds that Cesare is one of those people that control a computer, then he slowly tries to learn what is going on as he tries to come to grips with what he thinks about all this since he is a god fearing lad. Eventually the alure of forbidden knowledge and curiosity take over and Julius start to secretly studying Cesare's work from the few glimpses he can catch.
The problem on which Cesare is working is of central importance because it also serves to illustrate the power of owning a computer and the role of computer simulations in the scientific discovery process, which is something I would like the readers to experience.
The work is framed in the 1800s. A time of religious fervor and intellectual effervescence that will shake the world. In fact, this is also one of the themes of the work, the tension between tradition and innovation. So in order for the piece to have an interesting take I wanted them to deal with a problem that the scientific community of the 1800s could be aware with the physics of the time, but which was too hard to solve by hand, thereby requiring a computer and giving an advantage to those that owned one.
My first idea was the discovery of Neptune, but Urbain LeVerrier did the computations by hand and managed to get it published. So even though it might have been an extensive and difficult computation to perform, it was nonetheless quite possible with the computational resources of the time. I wanted something equally iconic and worldview shattering but harder, way harder. A juicy problem that is right there but is untractable without a computer.
What difficult, important problems could possibly be conceived with the knowledge of 1800s physics but were untractable without the employment of a digital computer's computational power?
I see a lot of great examples like Wheather forecasting and dynamical Many-body problems. However it ocurred to me that the examples (probably due to my lack of clarity in the question) are too general.
What do I mean by this? Well let's take the example of the N-Body problems for N>2, kindly suggested by @b.Lorenz, which is currently our best-ranked answer.
Although I can appreciate the difficulties to obtain analytical solutions to an N-Body problem and the contributions that having a computer would make to it, I was wondering what specific N-body problem would be relevant?b In the story, Julius and Cesare are posed to make a discovery using their computer. An important discovery. I wondered what that discovery might be. For instance, I mentioned the discovery of Neptune as my initial consideration.
This discovery could be a specific case of solving an N-body problem. The characters could simulate a system with a fictitious mass positioned so as to rid the predictions from inconsistencies with the observations, and then extrapolate the system into the future to make an observation of the hidden planet. That would be awesome. But I also mentioned that Urbain LeVerrier made a ton of approximations, calculated the thing by hand using data collected from astronomical observations and still got it right. Neptune was observed in the sky roughly around where LeVerrier predicted. So this particular problem of an N-body computation is not ideal because it is not a problem that highlights the advantages of computer simulation, it is not out of reach for a dedicated investigator with a few assistants working for, say, a year or a few months.
So what would be a good example then? What particular case of an N-body phenomenon would be relevant and known for 1800s physicists that they just wouldn't be able to investigate even if they managed to have the relevant insight?
Likewise, what specific case of wheather forecast or hurricane prediction would be dramatic enough that it would shatter the scientific community when solved, is knowable from the physics of the day, but is out of reach for the actual solution even to the brilliant minds that happened to stumble upon it? I know that "hurricane prediction" would be an incredibly useful but I'm thinking more on a specific instance.
So again I have noticed that I have a somewhat lack of clarity. AlexP asked what do I mean by "solve". Admitedly, solving a problem might mean a lot of things, particularly in physics.
Although I accept answers that deviate from this norm, if they are good and interesting, by "solving" I mean:
"Numerically integrating difficult systems of differential equations in order to obtain accurate numerical predictions for phenomena that, at the time of the setting, cannot be analytically obtained, that is by algebraically solving the system."
I require that the phenomenon in question be, if not useful or critical for the entirety of the society, at least scientifically interesting. For instance, the discovery of Neptune did not spell doom for the society, nor would it have suffered dire consequences if the planet had not been discovered, nevertheless it was important because it basically rewrote what we knew about our cosmic home, and that was a big deal for the astronomy of that time.
As I said, I will accept things that are not part of the above set, as long as they are interesting and enticing, but I believe that numerical integration of difficult systems of equations is a nice, basic aim.