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We present an algorithm for the numerical solution of ordinary differential equations by random enumeration of the Butcher trees used in the implementation of the Runge-Kutta method. Our Monte Carlo scheme allows for the direct numerical evaluation of an ODE solution at any given time within a certain interval, without iteration through multiple time steps. In particular, this approach does not involve a discretization step size, and it does not require the truncation of Taylor series.
G. Penent and N. Privault. Numerical evaluation of ODE solutions by Monte Carlo enumeration of Butcher series. Preprint arXiv:2201.05998, to appear in BIT Numerical Mathematics, 2022.