How can we define the precision we require in a numerical differential equation solver? What is it that I have to optimize to know? And how do I know that I'm at a sufficient time-step value?
For example, in those programs like Mathematica and Matlab, how does the differenital equations solver know the suitable step-size for your problem? How could that be automatically determined?
A full explanation to the approach would be nice (or maybe a term that can be googled).
Thanks for any efforts.