# Accuracy of Runge-Kutta compared to Taylor expansion

Let's say I have an ODE of the form : $y'(x) = f(x,y(x))$. I've been told that using the Runge-Kutta method for solving this ODE is equivalent to using Taylor expansion if $f(x,y(x))$ is linear in $y$ and that otherwise, one method would not be as accurate as the other. Why is that? I do not understand why this condition guaranties the equivalence between the two method. Furthermore, it seems to me that Runge-Kutta is always equivalent to Taylor due to the fact that RK is simply based on Taylor. Also, if this is false which method then would be more accurate and why?

## 1 Answer

See Butcher trees, short summary by Butcher himself at https://www.math.auckland.ac.nz/~butcher/ODE-book-2008/Tutorials/.

The RK method has an error order that is determined by how many terms in the Taylor expansion of $$\frac{y(x+h)-y(x)}{h}-\Phi(x,y(x),h)$$ vanish, where $y(x)$ is some exact solution and $\Phi$ is the summarized RK step, as in $$y_{n+1}=y_n+h·\Phi(x_n,y_n,h)$$.