用Butcher级数的Monte Carlo enumeration方法数值计算ODE解
Numerical evaluation of ODE solutions by Monte Carlo enumeration of Butcher series
摘要：
本文提出了一种通过随机枚举用于实现龙格-库塔法的Butcher树来数值解常微分方程的算法。我们的Monte Carlo方案允许在一定的时间间隔内的任何给定的时间，通过多个时间步长迭代的ODE解决方案的直接数值评估。特别地，该方法不涉及离散化步长，并且不需要截断泰勒级数。
英文摘要：
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.
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