# How can one describe the accuracy of a Runge-Kutta method?

I am solving a nonlinear ODE with a regular singularity using MATLAB ODE45 or ODE113.

I am wondering what precision and accuracy they have and what one can say about the numerical error. The idea would be to write a statement like

"The solution was found by means of ODE45 which has an accuracy of...".

• ODE113 does not use a Runge-Kutta method. Nov 1, 2015 at 4:34
• I know that but thanks for pointing that out. I could have written that it uses Adams-Bashforth-Moulton but the question about the accuracy would still apply to it... Nov 1, 2015 at 12:40
• What I meant to imply is that the title of your question does not accurately reflect the content. Nov 2, 2015 at 6:13

The RK4* method is a fourth-order method, meaning that the local truncation error is on the order of $O(h^5)$, while the total accumulated error is order $O(h^4)$.
• NB: Dmitry's statement is valid only in the asymptotic convergence regime $h \rightarrow 0$ (i.e., when $h$ is sufficiently small). Oct 30, 2015 at 13:41
• In the most widely accepted terminology, the local truncation error of this method is actually $O(h^4)$. The one-step error is $O(h^5)$. Nov 1, 2015 at 4:33
Since ODE45 uses adaptive stepping, you might want to mention the error tolerance (RelTol and AbsTol) used. There is some information in the odeset documentation.