What usecase do numerical methods for ODEs with orders of consistency of over 10 have in real world applications.
Since matlabs standart solver is a RK54 I wonder what problem necessitates a solver with such high consistency.
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You do know about the relation between order, step-size and error? See for instance https://math.stackexchange.com/a/1239002/115115 for an exploration of how the increase in the order reduces the number of function evaluations necessary for a given error level. This is for fixed-step iterations, but a similar pattern also applies to controlled step sizes. See "dense output" on how to intelligently construct values between the step points, which might be desired in the latter case. – Lutz Lehmann Feb 05 '21 at 17:17
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I am aware of those relations, yes. I am interested to know, if there are problems that necessitate methods with such a high order, for example because the required step sizes for other solving methods (e.g. Heun, RK4) would be so low, that they become in practical – lookatdatcake Feb 05 '21 at 17:21
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Astronomy, that is, solar system simulations etc., uses high-order solvers for its long-term simulations. You could look up if you can find the papers of Dormand-Prince or Verner where they develop their high-order methods, and what background they develop there. – Lutz Lehmann Feb 05 '21 at 17:57
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Cool, thank you for the answer, I will look into that. – lookatdatcake Feb 05 '21 at 18:05
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Some discussions of a very similar topic by several Runge-Kutta artists: https://scicomp.stackexchange.com/questions/25581/why-are-higher-order-runge-kutta-methods-not-used-more-often, also one can get an impression on the importance of order from the remarks in http://www.stochasticlifestyle.com/comparison-differential-equation-solver-suites-matlab-r-julia-python-c-fortran/ – Lutz Lehmann Feb 05 '21 at 18:29