EDIT: The answers below were helpful, but didn't get at the core of the problem, which I think has to do with the impulse aspect of things. I've since re-branded this question at Runge Kutta with Impulse . Cheers.
So I'm developing a method for studying ODEs that is similar in style to 'impulse response' but which is more general, though I have yet to find something as analytically beautiful as a Greens Function for it. Right now, everything is numerical/computational.
Having never done this before, I basically copy and pasted Runge-Kutta equations from Wikipedia into my simple python script. The results look beautiful, but I'm concerned that there are inconsistencies. Basically I'm trying to measure something I call 'energy residence time' using simulations of the ODEs, but I get different results for different step sizes and I'm not sure things are converging at smaller step sizes. Are there standard ways I can test my code/algorithm to make sure its legit, and hone in on where my problems lie?
If you want more info on the method itself (and the notion of energy residence time), I'd love to discuss!