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Here is the formula for determining order of convergence, $q$ is the order of convergence if we can find a constant $\mu$ that the fraction converges to as $k \to \inf$...

$$\lim_{k\to \infty} \frac{|x_{k+1}-L|}{|x_k-L|^q} = \mu \,\big|\; \mu > 0.$$

I know that this formula applies to root finding methods such as Newtons method or the Secant method where we can always take another iteration and find a value closer to the root. So we can let the number of iterations taken, $k$, approach infinity.

But does this formula apply when it comes to initial value problems and we are using Eulers method or Runge Kutta methods? As we set a fixed number of iterations dependent on the step size and the location $x$ where we want to find the value of the function. So if $x_0 = 0$, we want to find the value at $x = 4$ and we have a step size of $0.5$ we will have $k = 8$ iterations. We can't just 'take another iteration' and find a value closer, the number of iterations are set.

So in that case does the formula above not apply to initial value problems?

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up vote 1 down vote accepted

The given criteria is also valid for determining the order of a Runge-Kutta Method (or Euler Method). As you already mentioned you can not fix $h$ and let $k \rightarrow \infty$. As $h \propto k^{-1}$ you get $h\rightarrow 0$ as $k \rightarrow \infty$. But this isn't really a problem, as you can't fix the stepsize while increasing $k$ in i.e. numerical integration either.


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