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Suppose that we want to numerically solve the initial value problem $x'(t)=f(x,t), x(0)=x_0$. The modified Euler's method $$x(t+h)=x(t)+hf(t+\frac{1}{2} h,x(t)+\frac{1}{2} hf(t,x(t)))$$

My question, in fact this question from Kincaid and Cheney's book, how to use Richardson extrapolation on Euler's method with step size $h$ and $h/2$ in order to derive the Modified Euler method?

I can improve the method with simply saying, Euler method is $x(t+h)=x(t)+hx'(t)+Kh^2$ then using method of Richardson extrapolation for general methods but i don't see how to get modified Eulers.


Now these are What I think

Since $x'(t)\approx\frac{1}{h}(x(t+h)-x(t))$, so say $L = x'(t)$ and $\phi(h) = \frac{1}{h}(x(t+h)-x(t))$ and we have

$L = \phi(h)+hK+O(h^{2})$ and $2L = 2\phi(\frac{h}{2})+2K\frac{h}{2}+O(h^{2})$ then substraction second from the first we have $$L= 2\phi(\frac{h}{2})-\phi(h)+O(h^{2})$$

Now the question is that how can we show that $$2\phi(\frac{h}{2})-\phi(h) = f(t+\frac{1}{2} h,x(t)+\frac{1}{2} hf(t,x(t)))$$ Still have no clue,

Thanks for help and hints

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You should see this. –  J. M. Dec 26 '11 at 23:51
    
Thanks for the link, I looked over the link you suggested, i have to read Richardson extrapolation one more time. –  Aranel Dec 28 '11 at 3:07
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