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When does an ODE integrator reach the best numerical accuracy?

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I'm using an ODE Matlab integrator (Bulirsch-Stoer) that is (supposedly, I never used it before) quite good for solving ODEs with high accuracy. I'm using it to solve highly non-linear diff. eqs. of which one does not know the true solution and which are highly senstivie to initial conditions and numerical accuracy. To be sure that it does the right thing I integrate the curves back and forth and see whether the back-integration returns to the intial conditions. It does. Of course, if the time interval (the ODE's time-parameter "t") is large enough one sees it diverging (i.e., it doesn't return to the intital conditions.) This is normal, and expected. However, since the integrator's accuracy also depends on three independent parameters (the error tolerance, and two internal loops, midpoint and nr. of segmentation points) and it is difficult to find the optimal setting in a large 3D parameter space, I'm wondering whether the divergence from a true solution at some time, say t_max, beyond which the integrator no longer funrishes the correct values, is due to a non-optimal setting of the accuracy parameters (that is, I could still increase the accuracy), or wheteher it has reached the precision that is limited by the internal 15 digits number representation (that is, I reached the max. accuracy, and can't do anything about it as long I work with double precision.)
So, my question is: Is there a method to know when one has effectively reached the best numerical evaluation in solving an ODE inside the limitation of a double precision integratioon? A method that essentially tells me: "Yes, that's the best integration, beyond which you can't go with a 15 digits internal representation, no matter how you fine-tune your solver."
I hope that I expressed clearly what my issue is. Feel free to ask for more information.
  4 Comments
Steven Lord
Steven Lord on 26 Aug 2024
Over what time scale are you hoping to integrate? If you're trying to model that system over the span of years or even days, my guess is you're going to be out of luck if the system is that sensitive.
Marco
Marco on 26 Aug 2024
In the literature I found people integrating up to 200 seconds.Would like to get there to.

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Answers (1)

Torsten
Torsten on 26 Aug 2024
Edited: Torsten on 26 Aug 2024
Solve the problem for different error tolerances and see when the solution stabilizes.
If available, also test different solvers for the problem.
That's all you can do.
Some problems are irreversable. So you shouldn't use the back-integration method you suggested as a test for a successful integration.
  6 Comments
Torsten
Torsten on 26 Aug 2024
Hard to check if the noise is introduced by the solution method itself and you don't have a reference solution.

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