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I've come across multiple assignments now where I am to calculate trajectories, orbits etc. using the Runge-Kutta 4th order method. However, each time I am also asked to perform the correct convergence tests to make sure that I have a valid time step for my method.

What exactly am I looking for when performing the convergence test? And how would I do this for numerical methods where I can't compare with analytic results?

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    $\begingroup$ This question would probably be a better fit for computational science SE $\endgroup$
    – By Symmetry
    Commented Sep 19, 2018 at 14:37
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    $\begingroup$ If resources are not limited, you can just run the algorithm twice with a different step size. The $L^1$ distance (or any other norm) between the solutions is a good estimate of the error. There are much better approaches, but this one is conceptually very simple. $\endgroup$
    – AccidentalFourierTransform
    Commented Sep 19, 2018 at 14:41
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    $\begingroup$ I'm voting to close this question as off-topic because it belongs on scicomp.stackexchange. $\endgroup$
    – user4552
    Commented Sep 19, 2018 at 15:31
  • $\begingroup$ Orbits with RK4!? You may want tolook into symplectic integrators. $\endgroup$
    – Kyle Kanos
    Commented Sep 19, 2018 at 18:36
  • $\begingroup$ I discuss convergence wrt PDEs in this question. It may be of use to your situation (but not a dupe). $\endgroup$
    – Kyle Kanos
    Commented Sep 19, 2018 at 18:38

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