An explicit 16-stage Runge–Kutta method of order 10 discovered by numerical search

This article presents the discovery of an explicit 16-stage Runge–Kutta method that numerically satisfies the Runge–Kutta order conditions (in Butcher form) to 10th order, conjecturally improving the best-known number of stages in an explicit 10th-order Runge–Kutta method from 17 to 16. Unlike the vast majority of published Runge–Kutta methods, the method presented in this paper was not constructed via symbolic analysis or the use of simplifying assumptions, but instead by directly applying numerical optimization and root-finding algorithms to the order conditions. A naïve implementation of these algorithms would be made computationally infeasible by the considerable size and complex structure of the Butcher equations. However, we present a collection of software optimizations that greatly accelerate the evaluation of the Butcher equations and their derivatives while mitigating the effects of destructive cancellation, pushing these techniques within the realm of feasibility. While we do not have a formal proof of order, we present the results of numerical experiments to demonstrate that our method satisfies the Butcher equations to an accuracy of over 3000 decimal digits.