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I'm students at undergraduate school 2year

I don't have much background knowledge

I study nummerical analysis Runge - Kutta Method not a few

my text book induce Runge - Kutta midpoint method by 2 order Taylor series $T^2$ and it's approximating function $a_1f(t+\alpha_1,y+\beta_1)$

this approximation error is $O(h^2)$

since $w_{i+1}=w_i + h T^2(t_i,w_i) $ where $T^2(t,y)$ is order 2 Taylor for f(t,y). its local truncation error is also two

then replace $ T^2(t_i,w_i)$ to $ f(t_i+\frac{h}{2},w_i+\frac{h}{2}f(t_i,w_i))$ thus $w_{i+1}=w_i + hf(t_i+\frac{h}{2},w_i+\frac{h}{2}f(t_i,w_i))$

is this local truncation error also two because approximation error and local truncation error of $w_{i+1}=w_i + h T^2(t_i,w_i) $ is same for two?

is it correct?

sorry to my poor English ability

seyunkim
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  • Local truncation error is a technical term: it means the error made by taking a Taylor approximation over one time step. – Ian Jul 24 '19 at 15:05
  • (Cont.) "Approximation error" is not a completely technical term. My guess is that it means "global error". In general global error is one order worse than local truncation error. That's because the global error is basically the sum of all the local truncation errors, but there are $O(1/h)$ of those on a finite time interval. In the full story there is a bit more to it, which is where stability of the discrete method comes into play; basically the above is the story if the discrete method is stable. – Ian Jul 24 '19 at 15:05
  • then why $w_{i+1}=w_i+hf(t_i+\frac{h}{2},w_i+\frac{h}{2}f(t_i,w_i))$ is local truncation error two? – seyunkim Jul 24 '19 at 15:10
  • If I recall correctly, that method is "second order accurate", which means that the global error (assuming stability) is $O(h^2)$ and the local error (of approximating the overall expression $hf(t,y)$) is $O(h^3)$. I should emphasize this point: the exponent depends on how you group terms together. For example, a $O(h^2)$ error in approximating $f(t,y)$ produces a $O(h^3)$ error in approximating $hf(t,y)$ for example. – Ian Jul 24 '19 at 15:26
  • my text book say that this truncation error of mid point method is $ O(h^2) $ – seyunkim Jul 24 '19 at 15:34
  • That can make sense, but you need to pay careful attention to how exactly they define "local truncation error". If it is the error in approximating $y(t_{n+1})$ given exact knowledge of $y(t_n)$, then it should be $O(h^3)$. But if it is the error in approximating $f(t_n,y(t_n))$, then it should be $O(h^2)$. – Ian Jul 24 '19 at 15:34

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