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I'm looking for adaptive controllers (adaptive in both step size and order) for stiff integrators. I have asymptotically correct error estimates for the current method and all candidate methods of order 1 higher and lower than the current method. My naive controllers have occasional problems with either oscillating between different methods despite smooth long-term behavior, or getting stuck (e.g. with a high order method and unreasonably short time steps).

For the curious, these are IRKS general linear methods, see Butcher, Jackiewicz, and Wright 2007.

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Out of curiosity, is there any reason why you haven't tried extrapolative methods (e.g. the one by Bader and Deuflhard)? As I seem to recall the current state of the art has gotten pretty good in adjusting step size and order as appropriate. – J. M. Aug 13 '10 at 15:29
up vote 2 down vote accepted

Have a look at chapter 8 of Jackiewicz's book, especially section 8.10 for a general background. There's some matlab code by Podhaisky too, used to do this, but no control here.

And then, the theses of Butcher's recent student are here, which discuss implementation details, in particular Huang's chapter 3 should be very useful to you.

Older fortran code by Hairer do implement both order and stepsize control: see RADAU and DR_RADAU here, it's not for IRKS but gives a well-tested framework that could be suitably modified.

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Thanks, I wasn't aware of Jackiewicz's book or Huang's thesis. The "atlas" tables have a couple bugs, but I've used irks.m as a reference for computing coefficients. I had looked at Hairer & Wanner's RADAU a long time ago, but the error estimates have a significantly different form, so I think I concluded that it was not of significant value at the time. – Jed Aug 6 '10 at 16:23

The paper "Automatic control and adaptive time-stepping" by G. S\"oderlind might be useful. It deals mainly with stepsize control.

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If I may, I would also like to mention this predecessor paper by Gustafsson on controlling implicit RK methods: . – J. M. Aug 6 '10 at 4:11
Thanks, I'm aware of Söderlind's paper (and a 2006 paper in which he mentions that the methods are not expected to work well for DAE), it unfortunately discusses only step size control (not order). The framework is certainly capable of more general problems, but I would very much like to see it's performance on some more difficult problems. – Jed Aug 6 '10 at 16:40

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