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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...".

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    $\begingroup$ ODE113 does not use a Runge-Kutta method. $\endgroup$
    – David Ketcheson
    Nov 1 '15 at 4:34
  • $\begingroup$ 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... $\endgroup$
    – Hamurabi
    Nov 1 '15 at 12:40
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    $\begingroup$ What I meant to imply is that the title of your question does not accurately reflect the content. $\endgroup$
    – David Ketcheson
    Nov 2 '15 at 6:13
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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 $O(h^4)$.

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  • $\begingroup$ Thanks @Dmitry. Do you think that for a second order ODE with nonlinearity and regular singularity Runge-Kutta yields trustworthy results? $\endgroup$
    – Hamurabi
    Oct 30 '15 at 13:15
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    $\begingroup$ NB: Dmitry's statement is valid only in the asymptotic convergence regime $h \rightarrow 0$ (i.e., when $h$ is sufficiently small). $\endgroup$
    – GoHokies
    Oct 30 '15 at 13:41
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    $\begingroup$ In the most widely accepted terminology, the local truncation error of this method is actually $O(h^4)$. The one-step error is $O(h^5)$. $\endgroup$
    – David Ketcheson
    Nov 1 '15 at 4:33
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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.

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