I'm a little unsure if there are general rules for which method to use in an adaptive ODE scheme that uses two methods of different accuracies to estimate the error.
For example, the approach I am used to is estimating the one step error of the lower order method. Then what I have read is that the lower order method is used to advance the estimate because that is the error we actually bounded. But on the flip side, we generally would assume the higher order method will be more accurate, so it still seems reasonable to advance the estimate with the higher order method. Is there a general rule/argument to follow?
If helpful the argument I have seen, in short is that if $w_i$ and $\bar{w}_i$ are the the lower and higher order estimates respectively of the exact solution $y_{(t_i)}$, then we can bound the relative one step error of the lower order method as
$$ \frac{|w_i-y_{(t_i)}|}{h_i} \lessapprox \frac{|w_i-\bar{w}_i|}{h_i}$$
This in part is coming up for me because of the wikipedia page on the Dormand-Prince method https://en.wikipedia.org/wiki/Dormand%E2%80%93Prince_method. It says that the coefficients were chosen to minimize the error of the higher order method. It goes on to say "This is the main difference with the Fehlberg method, which was constructed so that the fourth-order solution has a small error".
To put my question a different way, if the adaptive scheme (no matter the underlying methods) is being implemented like the above where the one step error of the lower order method is being estimated, is it best practice to always update with the lower scheme? Then I am assuming there are other approaches that bound the error of the higher order scheme and in those cases the higher order estimate should be kept?