Fourth order Runge–Kutta in C# - two differential equations

[Itosu]

So if I get:

h AUC(after 2 hours)
0.01 335.40257585727842
0.005 335.87803342989893

Can I presume that 335 is accurate result? (without knowing the part after comma).
If h=0.01 does O(h⁵) mean that up to the 0.01⁵ I get accurate digits?

And should I count O(h⁵) or O(h⁴)? Local/global error?

 

[I like Serena]

If h=0.01 does O(h⁵) mean that up to the 0.01⁵ I get accurate digits?

It means that up to *a constant times* 0.01⁵ you will get accurate digits.
This constant is unknown (and might be bigger than you think or suspect).

So if I get:

h AUC(after 2 hours)
0.01 335.40257585727842
0.005 335.87803342989893

Can I presume that 335 is accurate result? (without knowing the part after comma).

Since the constant is unknown and you have a discontinuity in your function, you'll have to experiment a bit.
Try a step like 0.0025.
I expect that you'll see an AUC that starts with 335.8, and probably even 335.87 or 335.88.
That should give you "confidence" that 335 are indeed accurate digits, but there is no "guarantee" mathematically speaking.
The result to present, based on your calculation, would be 335.9 ± 0.5 or 335.88 ± 0.48.
Note that whenever you measure something and present a result, you are supposed to always include the margin of error in your result.

 

[Itosu]

Thank you, I understand.
Could you explain more deeply the idea of the error O(h⁵) or provide a link?

Thank you in advance!

 

[I like Serena]

Thank you, I understand.
Could you explain more deeply the idea of the error O(h⁵) or provide a link?

Thank you in advance!

The errors of numerical algorithms are expressed with this big O notation:
http://en.wikipedia.org/wiki/Big_O_notation"