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I have a system of nonlinear first-order autonomous IVP ordinary differential equations for which I'll solve numerically since I can't obtain a closed-form solution.

What are the notions that matters most when comparing candidate numerical methods in this particular case? Will I be able to obtain a more refined answer given the system at hand?

What are the tools used to quantitatively estimate the errors (locally and globally)?

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    It's a bit rough. If you don't need that much accuracy and the right-hand sides are cheap to evaluate, Runge-Kutta works nicely. If you need slightly more accuracy and the right-hand sides can be evaluated cheaply, extrapolative (Gragg-Bulirsch-Stoer) methods should be considered. If the right-hand sides are expensive to evaluate, multistep (Adams) methods might be appropriate. Here I am assuming that the problem isn't stiff. A good nonstiff solver ought to return a warning if what it's solving seems stiff...2011-09-23
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    ...and all the methods I gave have counterparts for stiff equations. Any good routine should have some form of error control/estimation built in. It might be more helpful if you post the exact system you have, and the typical ranges of parameters/initial values you're treating.2011-09-23
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    Finally: if what you have is in fact a differential-algebraic equation (DAE), then that's a different can of worms.2011-09-23
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    @J.M. Thank you very much, I'll keep these rule of thumbs at hand. I have a non-stiff system. What I was interested in was rather understanding how an estimation of error would go.2011-09-24
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    Ah, then you'll want to see [this paper](http://dx.doi.org/10.1007/BF01389440). Well actually, read [Hairer/Norsett/Wanner](http://books.google.com/books?id=F93u7VcSRyYC) first, and then you can look at the papers...2011-09-25
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    @J.M. Great, thanks.2011-09-25

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