When does an ODE integrator reach the best numerical accuracy?

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Marco 2024-8-26，6:47

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https://ww2.mathworks.cn/matlabcentral/answers/2147899-when-does-an-ode-integrator-reach-the-best-numerical-accuracy

评论： Torsten 2024-8-26，20:33

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.

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Steven Lord 2024-8-26，16:38

What is the problem you're trying to solve that requires a down-to-the-last bit solution? Are you solving an orbital mechanics problem over the course of years or decades? If we know the specific type of problem you're trying to solve, I'm wondering if there's a domain-specific tool or algorithm that can solve that problem more easily or effectively than a general purpose ODE solver. In a pinch you could use a wrench to hammer in a nail, but you're probably going to be more effective if you use a hammer instead.

If you require an exact solution you could always go to a symbolic solution (depending on whether or not your system of ODEs has an analytical solution) but that likely will take more time if it is possible.

Are you looking for a theoretical best or an actual best approach (based on the specific problem or type of problem you're trying to solve)?

In a comment on Torsten's answer you wrote "As to the irreversibility, I'm dealing with ODEs (y'=f(y,t)) having a unique solution y(t). Integrating it from y(0) to y(T) with T>0 must coincide with the same integration from y(T) to y(0)." Is that "must" in theory, in practice, or both? As the old saying says, "In theory there is no difference between theory and practice. In practice there is."

Steven Lord 2024-8-26，18:15

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 2024-8-26，18:45