Numerical evaluation of ODE solutions by Monte Carlo enumeration of Butcher series

TL;DR

This paper introduces a Monte Carlo algorithm for solving ODEs by randomly enumerating Butcher trees, enabling direct evaluation at specific times without traditional discretization or Taylor series truncation.

Contribution

It presents a novel Monte Carlo method that directly computes ODE solutions at arbitrary points, bypassing step-by-step iteration and discretization.

Findings

Enables direct evaluation of ODE solutions at any time point

Eliminates the need for step size discretization

Avoids Taylor series truncation in numerical solutions

Abstract

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.