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I'm trying to solve a series of problems related to approximations of ODEs with Runge-Kutta that have their approximation to values improved by using the Richardson Extrapolation.

Some of these problems propose the use of step sizes different than $h/2$ when comparing the solution obtained with a step-size $h$. For example, one of the problems asks me to apply Richardson Extrapolation with two steps: one of $0.25h$ and then another of $0.75h$, which then are to be compared with one of the $h$ step iterations.

I've been given the error formula $\epsilon= 2^r \frac{y^\frac{h}{2}_{m+1} - y^h_{m+1}}{1 - 2^r}$ to calculate the error for an iteration of step-size $h$ when applying Richadson Extrapolation with a secondary step-size of $h/2$, with $r$ being the order of the used method and $m+1$ representing the number of the given iteration.

I'm completely lost in finding out how to adapt that formula for the $0.25h$ and $0.75h$ step-sizes or how to derive the formula starting from the Taylor formula.

Please help. I've already used the search and it doesn't seem to be a problem of this specific kind already posted in Math.StackExchange.

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1 Answer 1

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The underlying idea is that locally the step-size-error dependence is fixed and can be expanded as power series. Further that errors are, in first approximation, additive.

This means that the error of step size $h$ is $ch^{p+1}+O(h^{p+2})$, $p$ the order of the method, and the errors of the partial steps are thus scaled $ch_k^{p+1}+O(h_k^{p+2})$. This gives $$ y_1=y_*+ch^{p+1}+O(h^{p+2})\\ y_2=y_*+ch^{p+1}[(\tfrac14)^{p+1}+(\tfrac34)^{p+1}]+O(h^{p+2}) $$ Now the extrapolated value is obtained by eliminating $ch^{p+1}$, that is $$ [(\tfrac14)^{p+1}+(\tfrac34)^{p+1}]y_1-y_2=[(\tfrac14)^{p+1}+(\tfrac34)^{p+1}-1]y_* \\~\\ y_*=\frac{4^{p+1}y_2-[3^{p+1}+1]y_1}{4^{p+1}-[3^{p+1}+1]} $$

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