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

TL;DR

This paper develops higher order symmetric partitioned Runge-Kutta methods for differential equations on Lie groups, demonstrating they allow larger step sizes and have predictable error properties, improving numerical simulations in lattice QCD.

Contribution

It introduces time-reversible higher order integrators based on implicit partitioned Runge-Kutta schemes for Lie-group equations, analyzing their error and convergence properties.

Findings

SPRk schemes permit larger step sizes than Leapfrog.

Global error of SPRK schemes is always even, affecting convergence order.

Methods based on exponential series truncation achieve predictable accuracy.

Abstract

Computer simulations in QCD are based on the discretization of the theory on a Euclidean lattice. To compute the mean value of an observable, usually the Hybrid Monte Carlo method is applied. Here equations of motion, derived from an Hamiltonian, have to be solved numerically. Commonly the Leapfrog (Stoermer-Verlet) method or splitting methods with multiple timescales \`a la Sexton-Weingarten are used to integrate the dynamical system, defined on a Lie group. Here we formulate time-reversible higher order integrators based on implicit partitioned Runge-Kutta schemes and show that they allow for larger step-sizes than the Leapfrog method. Since these methods are based on an infinite series of exponential functions, we concentrate on the truncation of this series with respect to the global error and accuracy. Finally, we see that the global error of a SPRK scheme is always even such that a convergence order of one is gained for methods with odd convergence order.