Discretization of Continuous Mathematics

I am currently taking a course involving the use of numerical methods to solve partial differential equations. I have not yet been exposed to such a technique and as an aspiring computer scientist, am particularly intrigued as to its implications. Specifically, can anyone speak to how pervasive the discretization of continuous mathematics is? Can anyone recommend further reading on the subject? I am particularly interested in whether or not it would be possible to redefine fundamental physical laws, previously described using continuous mathematics, in terms of discrete mathematics.

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This can bite you, especially early in your career, in very surprising ways. For instance, I once wrote a short graphical program in which the camera would abruptly point straight up, with no remedy except restarting the program. It turned out that all of my camera rotations were very slightly reducing the length of the camera's "look" vector, and it eventually became $\vec{0}$.