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During trying to solve a differential equation using numerical methods like RK methods (Runge Kutta) , one has to factor in important stuff like stability of the the system , whether the equations are stiff or not. For that , appropriate step size has to be considered.

However, as I have read that equations becomes stiff due transient spikes due to which step-size need to be smaller. However, I am aware of schemes like Adaptive RK Method where we can move along x , adjusting the step size as per the local error per step. Why can't this step control in Adaptive RK Method be used to account in stability itself ?

Quoting the book , "Stability requirement will still necessitate using very small steps throughout the entire solution".

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With fixed step size you will just get a huge factor in the $O(h^4)$ error for stiff but smooth systems. If you have a method of order $p$, but the ODE function is truly only $C^{s,1}$ with $s<p$, then the resulting error order will be $O(h^{s+1})$. An order 2 method like Heun with a kink or fold in the ODE will, due to traversing the kink, have an error of $O(h)$.

With step-size adaption methods, the guessing of the local contribution to the global error depends on the derivatives of order $p$ (or $p+1$?) to exist and be continuous. If that is not the case, then the adaptation operates in the dark and will only make cautious small steps. However, since there is a lower boundary for useful step-sizes, this walking in the dark also has a limit.

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