Discovering New Runge-Kutta Methods Using Unstructured Numerical Search

Abstract: Runge-Kutta methods are a popular class of numerical methods for solving ordinary differential equations. Every Runge-Kutta method is characterized by two basic parameters: its order, which measures the accuracy of the solution it produces, and its number of stages, which measures the amount of computational work it requires. The primary goal in constructing Runge-Kutta methods is to maximize order using a minimum number of stages. However, high-order Runge-Kutta methods are difficult to construct because their parameters must satisfy an exponentially large system of polynomial equations. This paper presents the first known 10th-order Runge-Kutta method with only 16 stages, breaking a 40-year standing record for the number of stages required to achieve 10th-order accuracy. It also discusses the tools and techniques that enabled the discovery of this method using a straightforward numerical search.