You're being way too polite in your wording, as your next paragraph states, every field that uses computers has done it since the computers were invented. Allow me to push it further: AFAIC almost all hand calculated math is discrete. The numbers you can write on a page are countable, in fact finite, so all actually calculated math is "discrete" in some sense. No one every really touches the full continuum of R other than abstractly: real calculations whenever you truncate pi or sqrt(2), or generally calculate with a fixed set of digits, you are doing _discrete_math_. So yes, Q is dense in R which helps, and you could write out more digits if you really wanted to, but even then, people restrict themselves to small, finite subsets of Q and even resorting to using things like logarithms/order of magnitude to keep that set as small as possible since our brains can't take all the hairiness. But still, that set is large enough to still do the things we care about anyway.
However, when people want to think abstractly rather than explicitly (calculating), it's easier to take the limit as the gaps go to zero and deal with clean, C^\infty functions. I mean, that's literally what people mean (at least how my experimental physicist brain thinks it does) when they say analysis/calculus is about approximations. I deal with plasma in the hot, nonquantum limit so my ions and electrons are in fact discrete particles. Whenever I calculate an electric field or a magnetic field for the distribution, I ignore the fact that there will be noise on the scale of individual particles and replace that noisy function with a clean smooth function, which tends to be a good approximation. In fact, this is how almost every classical physics problem is solved, you ignore the structure on the particle level and pretend it is "continuous". Same with materials engineering, and so on.