the length of the time span in ode45

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!

Answer 1

[Jan]

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.

[Tony Cheng]

then I will set the timespan in ode45 as [ 0 0.001 ], rather than [ 0 0.1 ].

[Jan]

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.

Answer 2

[Torsten]

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.

[Tony Cheng]

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]

At t = 0.001: no.