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It seems that the 5th order is the corrector and the 4th order is the predictor. Can RK5 be considered as part of Predictor–corrector method?

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No. The 4th order is not a predictor for the 5th order. Instead, what is done is that the $k_i$ are constructed in such a way that one linear combination $b_i k_i$ gives a 5th order method and $\hat{b}_i k_i $ gives a 4th order method. They are calculated independently, but constructed in a way such that the exact same intermediate steps ($k_i$) build both methods, saving the computation of actually computing two RK methods at the same time. This is called method embedding.

Predictor-corrector is when one method is used as the input to the second (usually implicit method). That requires non-trivial calculation to compute the second.

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