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the length of the time span in ode45

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Tony Cheng
Tony Cheng il 24 Feb 2022
Commentato: Torsten il 24 Feb 2022
Hi there,
I am using ode45 to solve a set of ODEs. Here I want to ask, if the length of the time span is relevant to the accuracy of the numerical results? I mean, is there a relationship like, a longer timespan corresponds to a rough accuracy, or a shorter timespan corresponds to a higher accuracy? Many thanks!

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Jan
Jan il 24 Feb 2022
The small discretization and rounding errors accumulate over the time. Small deviations of the initial values can be amplified also. So the general rule is, that a longer integration time can cause larger deviations of the final value.
Of course a stable trajectory does not suffer from this: e.g. a falling pencil will be at the same position and some houres also, so increasing the time will not change anything.
This concerns the real world also: To determine the exact position of the earth in 1 million years, you need to know the current position with an extreme accuracy. But, by the way, even the "year" is not defined accurate enough for this job.
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Tony Cheng
Tony Cheng il 24 Feb 2022
then I will set the timespan in ode45 as [ 0 0.001 ], rather than [ 0 0.1 ].
Jan
Jan il 24 Feb 2022
Usually the length of the time span depends on the problem you want to solve. So if you need the trajectory from 0 to 1, using an earlier final time is not an option.

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Torsten
Torsten il 24 Feb 2022
If by "length of the time span" you mean" the number of output times (for the end time of integration fixed)", the answer is no - it is not relevant for the numerical accuracy of the numerical results.
  2 Commenti
Tony Cheng
Tony Cheng il 24 Feb 2022
Thanks for your reply!
What I mean is, for example, for the time span set in ode45, if I set the span as [ 0 0.001 ], will it produce better accuracy in the numerical results than that from [ 0 0.1 ]?
Torsten
Torsten il 24 Feb 2022
At t = 0.001: no.

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