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I understand that if we use a high-order Runge Kutta method like RK4 the rate of convergence of the error as the stepsize $h$ tends to $0$ should be of order $h^4$.

Does that necessarily imply that the global error (i.e. max norm of the difference between the exact and approximate solution) of say a fifth-order Runge Kutta method is lower than the error obtained with a fourth-order method? In other words, the error curve as a function of the stepsize of the fifth order method should be at least below the fourth order method, for all stepsizes?

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No, it is not necessarily so. The error of an $m$'th order method is bounded by (and usually approximately equal to) $C h^m$ for some constant $C$. However, different methods will have different values of $C$. As a result, although for sufficiently small step size $h$ the higher-order method should beat the lower-order method, for any particular $h$ there are no guarantees.

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