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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... –  J. M. Sep 23 '11 at 6:17
    
...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. –  J. M. Sep 23 '11 at 6:19
    
Finally: if what you have is in fact a differential-algebraic equation (DAE), then that's a different can of worms. –  J. M. Sep 23 '11 at 6:20
    
@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. –  Weaam Sep 24 '11 at 20:13
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Ah, then you'll want to see this paper. Well actually, read Hairer/Norsett/Wanner first, and then you can look at the papers... –  J. M. Sep 25 '11 at 0:50

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