Conserved quantities in the theory of discrete surfaces

Suppose you are given a simple first order smooth ordinary differential equation with a given initial condition. If you cannot write down its solution explicitly, you might find a discrete approximate solution by using the Euler or Runga-Kutta algorithm, just to have some initial idea how the smooth solution behaves. In this case, your interest in the approximate solution is only as a stepping stone for understanding the smooth true solution. We can think of the equation (i.e. the algorithm) for the discrete approximate solution as a finite dimensional problem because the full space of objects (a vector space of discrete functions) that can be inserted to test for validity in the equation is finite dimensional. Likewise, we can call the smooth differential equation an infinite dimensional problem (this might be somewhat unconventional), because the objects insertable into the equation form an infinite dimensional vector space.