Symmetric Partitioned Runge-Kutta Methods for Differential Equations on Lie Groups

TL;DR

This paper introduces a higher order symmetric partitioned Runge-Kutta method tailored for differential equations on Lie groups, enhancing accuracy and symmetry properties for complex simulations such as quantum field theories.

Contribution

It develops a novel symmetric partitioned Runge-Kutta scheme with convergence order 4 for differential equations on Lie groups, addressing symmetry issues at higher orders.

Findings

The SPRK method achieves convergence order 4.

Numerical comparison shows SPRK outperforms Störmer-Verlet in certain scenarios.

The method is applicable to quantum field theory simulations.

Abstract

In this paper, we develop a higher order symmetric partitioned Runge-Kutta method for a coupled system of differential equations on Lie groups. We start with a discussion on partitioned Runge-Kutta methods on Lie groups of arbitrary order. As symmetry is not met for higher orders, we generalize the method to a symmetric partitioned Runge-Kutta (SPRK) scheme. Furthermore, we derive a set of coefficients for convergence order 4. The SPRK integration method can be used, for example, in simulations of quantum field theories. Finally, we compare the new SPRK scheme numerically with the St\"ormer-Verlet scheme, one of the state-of-the-art schemes used in this subject.