How to choose the order of a Runge-Kutta method? I have seen that Runge-Kutta's methods are a family of methods used to approximate the solution of an initial value problem. I have also seen that they are classified depending on their order (with the second-order R-K being the Euler's Modified method, and the fourth-order R-K being the most used among them).
So, given a first-order ODE $y'=f(x,y)$ with an initial condition $y(x_0)=y_0$, what is the criteria to follow to choose the order of the Runge-Kutta method to be used?
 A: If you stay with explicit Runge-Kutta methods, then one can roughly say

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*order 1 to 3 are for educational use, to get an intuition about numerical ODE solvers.

*order 4, specifically the classical RK4 and the 3/8 method, are the high-end for fixed-step methods and good for producing plots of simple ODE

*for serious applications, step-size controlled methods should be used, usually based on embedded methods. Of these, the (4)5 methods (Fehlberg, Domand-Prince) are the most established ones, (7)8 or (8)9 methods seem to be optimal, that is, the known examples for orders 12 or 14 do not provide a better performance on current hardware.

This is for "proper" ODE systems that have mediums state dimension and tight coupling of the components. Simple rules usually lead to lots of exceptions, here for instance PDE discretizations and stiff ODE.

*

*For PDE discretizations according to the method-of-lines one often finds that the order in time direction is chosen equal or not much higher than the discretization order in space direction.


*For stiff ODE, that is, the adaptive step-size selection is actively limited by the stability region of explicit RK methods, implicit methods of slightly lower order can have a dramatically better performance in terms of ODE function evaluations for comparable error targets.
