how to determind the error of euler's method?

Question

I know that the formula is y''*h^2/2, where h is the distance between the interval. BUt what is the value of y''?

How do you determind the error bound of euler's method?

Help is needed, thanks.

Answer 1

under some conditions you can take for y''(s) the max value of the second derivative of y on each of the h-width intervals.
usually you dont know because that is the tunction you are trying to approximate.

in that case you can compare two solutions : one with stepwidth h and one width step width 2h and estimate the y'' by subtracting one solution from teh other


update :

The error you get in a point t0 is ( with one h- step
)

1/2 * h0^2 * y''(s) + O(h0^3)

where s is a point in the interval [t0, t0 + h0]

From your dv you need an upperbound for y'' on that
interval. You need to estimate the y'' find an upper
bounmd.

If you can not derive such an upper bound ( and you
generally cant ) then procceed as follows :

Do an iteration with h=h0/2
you will have then 2 approximations of y in t=t0:

approx1 : y(t0) + 1/2 * h0^2 * y''(s) + O(h^3)
approx2 : y(t0) + 1/2 * 1/4 * h0^2 * y''(s1) +
1/8*O(h0^3)

neglect the O(h^3)
combine both and you will be able to find an
upperbound for y'' ( for suffinciently small h )

---
Note that if you do several h-steps to arrive at t0
you will need to add up the errors but these will
still be of some contant * h^2
----

the real solution if a,b,c are contant of the dv is
known see for instance :
http://www.sosmath.com/diffeq/second/constantcof/constantcof.html