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I am currently reading about fourth order Runge-Kutta methods, and I noticed that for first order the total error is a factor of $O(h^2)$, and for fourth order the error is in order of $O(h^5)$, where $h$ is the step size. I understand that fourth order Runge-Kutta methods are better than first order since the error rate for each step is smaller, which only makes sense if $h < 1$.

My question is why must the step size be smaller than $1$? I've tried setting $h$ to something bigger than $1$ in my simulation and it exploded, but I'm not sure if that's a result of the approximation not working for step sizes bigger than $1$, or maybe my other coefficients were too large. So is there a requirement for $h$ to be smaller than $1$, and if so, why?

Alex M.
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WhiteTiger
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  • There's nothing magical about $1$; it just depends on the scale over which your function varies. The coefficients of the error terms will depend on the derivatives of the solution; that's what will really determine how small $h$ needs to be. – mjqxxxx Jan 07 '16 at 22:22
  • 1 what? 1 second, 1 hour, 1 year, 1 decade. It all depends on the problem at hand. The 1 by itself tell you nothing. – John Alexiou Jan 07 '16 at 22:57

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