# As step size $h$ decreases, which method is more efficient? Euler-midpoint method and the classical fourth-order Runge-Kutta method?

I came up with the following problem while using Euler-midpoint method and the classical fourth-order Runge-Kutta method to solve ordinary differential equations.

As step size $$h$$ decreases, which method is more efficient in estimation? Euler-midpoint method and the classical fourth-order Runge-Kutta method?

By solving several questions and comparing them to actual value given by exact solution, I realize that for the same step size $$h$$, the classical fourth-order Runge-Kutta method gives a more accurate estimation compared to the Euler-midpoint method.

However, does the same hold true as $$h$$ tends to $$0$$ from positive?

• This comes down to the errors of each method. The midpoint rule is known for $\mathcal O(h^2)$ local error (and is in fact a 2nd order RK method), while the classical RK 4 method has $\mathcal O(h^4)$ local error, which decays much faster. Commented Oct 17, 2020 at 13:42
• From the speed of decaying, as $h$ tends to $0$, one can conclude that RK4 method converges much faster than Euler midpoint method? Commented Oct 17, 2020 at 13:43
• Yes, $h^4$ vanishes faster than $h^2$. Commented Oct 17, 2020 at 13:45
• I see. Perhaps you can post an answer so that I can accept your answer? Commented Oct 17, 2020 at 13:45
• See math.stackexchange.com/a/1239002/115115 for an example. RK2 there is the explicit midpoint method. Note also the practical limitations of using floating-point arithmetic, there is no convergence for $h\to 0$ in the experiments, or rather one would have to switch to a variable-precision numerical data type. Commented Oct 17, 2020 at 14:07

This comes down to the errors of each method. The midpoint rule is known for $$\mathcal O(h^3)$$ local error or $$\mathcal O(h^2)$$ global error (and is in fact a 2nd order RK method), while the classical RK 4 method has $$\mathcal O(h^5)$$ local error or $$\mathcal O(h^4)$$ global error, which decay much faster.