It is definitely not obvious, but I wouldn't say it is completely unclear. For instance we know that algorithms like the leapfrog integrator not only approxima

It is definitely not obvious, but I wouldn't say it is completely unclear. For instance we know that algorithms like the leapfrog integrator not only approximate a physical system quite well but even conserve the energy, or rather a quantity that approximates the true energy. There are plenty of theorems about the accuracy and other properties of numerical algorithms.