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Use Richardson extrapolation to improve the formula $$ f''(x) \sim \frac{f(x+h)-2f(x)+f(x-h)}{h^2} $$ so that the error is reduced to order $h^4$

I am not sure how to go about doing this problem, if anyone could help it would be much appreciated, thank you for your help in advance.

user138294
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1 Answers1

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Richardson extrapolation is based on the fact that if the time-step is decreased to half then the error goes to $\frac{1}{2^p}$. The formula is second order accurate in $h$.

So we can write

$$ f''(x)=\frac{1}{h^2}\left(f(x+h) -2f(x) + f(x-h) \right) + Kh^2 + O(h^4) $$

Now we use half the step

$$ f''(x)=\frac{4}{h^2} \left(f(x+\frac{h}{2}) -2f(x) + f(x-\frac{h}{2})\right) + K\frac{h^2}{4} + O(h^4) $$

Multiply the second equation with $4$ and subtract the first to obtain

$$ f''(x)= \frac{1}{h^2} \left[ 16 \left\{f(x+\frac{h}{2})-2f(x) + f(x-\frac{h}{2})\right\} -\left\{f(x+h) -2f(x) + f(x-h)\right\} \right] + O(h^4) $$ EDIT: I had started incorrectly by assuming first order. The odd powers cancel out. Hope I got the rest of the math right.

yaska
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  • Thanks for pointing out the error @cheapeffectivedietpills Although, it did cross my mind that one wouldn't want to use it for improving by 3 orders. Tunnel vision is the greatest enemy of math, I suppose – yaska Mar 28 '14 at 08:10