# Numerically solving an ode with infinitely many variables of which only finitely many are significant in magnitude

Suppose I have an ode that involves infinitely many variables, with the property that at any given time, only finitely many of them are large enough to be of interest (say $$>10^{-10}$$). However, at different times, different variables may become large.

It is also the case that given such a set of interesting variables, only a finite number of equations contain terms that are large. This is somewhat like a generalized version of locality.

The question is, is there any research on solving such equations numerically? My idea is to keep track of the variables of interest, and also "secondary" variables, which are significantly (in magnitude) coupled with the "interesting" variables; we can also keep track of "tertiary" variables and so on. We then go on solving the ode, ignoring the uninteresting variables (assuming them to be 0), and check regularly if a new variable comes into (or goes out of) interest.

To give the background, I'm working on an artificial chemistry simulation. All the reactions, and their reaction rate formulae (following Arrhenius) can be determined by my set of rules. For example, $$A + X \to B\\ B \to X + C$$ simulates the conversion $$A \xrightarrow X C$$ with catalyst $$X$$. In this case, $$C$$ is initially small, but eventually becomes large. This system is finite, and so is solvable by conventional methods. But if the reations are infinite (but recursively enumerable and decidable, and for simplicity, each combination of reactants only result in a finite number or reactions), it creates an infinitely large set of ode.