Explicit and Effectively Symmetric Runge-Kutta Methods

TL;DR

This paper introduces Explicit and Effectively Symmetric (EES) Runge-Kutta schemes, which are explicit, near-symmetric methods that outperform some higher-order explicit schemes and match implicit symmetric schemes in efficiency.

Contribution

The paper develops a new class of explicit symmetric-like Runge-Kutta methods by minimizing antisymmetric components, enabling efficient integration of Hamiltonian systems and Neural ODEs.

Findings

EES schemes are second-order and outperform RK4 and RK5.

EES schemes achieve results comparable to implicit symmetric schemes.

EES schemes have significantly lower computational cost.

Abstract

Symmetry is a key property of numerical methods. The geometric properties of symmetric schemes make them an attractive option for integrating Hamiltonian systems, whilst their ability to exactly recover the initial condition without the need to store the entire solution trajectory makes them ideal for the efficient implementation of Neural ODEs. In this work, we present a Hopf algebraic approach to the study of symmetric B-series methods. We show that every B-series method can be written as the composition of a symmetric and "antisymmetric" component, and explore the structure of this decomposition for Runge-Kutta schemes. A major bottleneck of symmetric Runge-Kutta schemes is their implicit nature, which requires solving a nonlinear system at each step. By introducing a new set of order conditions which minimise the antisymmetric component of a scheme, we derive what we call Explicit and Effectively Symmetric (EES) schemes -- a new class of explicit Runge-Kutta schemes with near-symmetric properties. We present examples of second-order EES schemes and demonstrate that, despite their low order, these schemes readily outperform higher-order explicit schemes such as RK4 and RK5, and achieve results comparable to implicit symmetric schemes at a significantly lower computational cost.