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Could anyone suggest any good (perhaps online ref papers) reference material on numerical analysis focusing on determining accuracy/estimated errors, rates/orders of convergence especially when applied to ODEs? Thank you very much.

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Both of the books by Hairer and Wanner should be a good start. – J. M. Aug 21 '11 at 21:50
I think this should be converted to community wiki. – sxd Aug 21 '11 at 22:16
I like this book. It has two chapters on ODEs. – joriki Aug 21 '11 at 22:30
I would insert "with practical advice" in between "chapters" and "on" in @joriki's comment. – J. M. Aug 21 '11 at 22:36
@Dimitri: I disagree. As far as I know there aren't that many people here who could provide sound advice on numerical analysis, so I think it would be better to leave the question non-CW even if it might not have an objective correct answer. Reputation might serve as a little incitement to contribute a good answer. – t.b. Aug 22 '11 at 0:47

Modern ODE solvers do (or are supposed to) perform error monitoring and control in the course of evaluating a differential equation system given to it; what is done under the hood depends on which algorithm is being used. Here, I'll talk about nonstiff solvers (stiff solvers use similar ideas, but with some complications to account for the methods being more involved).

For Runge-Kutta solvers, it is now customary to use embedded (a.k.a. Fehlberg-type) Runge-Kutta formulae; briefly, an embedded RK method is any method that allows for the construction of at least two RK formulae of different orders of convergence from the same set of function evaluations; the difference between the results of the different estimates can then be taken to be the error of the approximation. These two technical reports be Fehlberg explain the idea, but Dormand and Prince and Bogacki and Shampine are two of the versions of embedded RK with better error control than Fehlberg's original methods.

For extrapolative (Bulirsch-Stoer) methods, you have in fact a sequence of estimates that (hopefully!) tend to the correct solution; Deuflhard's article demonstrates a practical way to estimate the error of a solution from an extrapolative scheme.

For multistep/multivalue methods, the situation is even more complicated, and highly depends on the particular implementation. I personally don't have much experience with tweaking a multistep code, so I can't say anything more.

Those are only for local estimates of the error; for global estimates, the situation is slightly more complicated. See this for instance.

You can probably tell after a casual glance of the papers I linked to above that this is a complicated business, and it is. So: read the Hairer/Wanner books I linked to in the comments for a start, and then have a look at these papers after.

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I have two preferred references: Atkinson's Introduction to Numerical Analysis and the excellent Numerical Recipes (freely available online - the section on ODE starts on page 899). I first learned from Atkinson, so I recall it very warmly. But it does, perhaps, leave much to be discovered in the exercises. Numerical Recipes is free, and does not.

In the comments, Numerical Methods that Work was mentioned, and I think this is a great book, too.

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Look through in the numerical methods documentation, and under physics samples.

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