Accurate and robust splitting methods for the generalized Langevin equation with a positive Prony series memory kernel

TL;DR

This paper introduces a new splitting method for the generalized Langevin equation with a positive Prony series memory kernel, significantly improving simulation accuracy and robustness across various parameters.

Contribution

The paper presents a novel, easy-to-implement splitting method that enhances the accuracy and robustness of GLE simulations with a positive Prony series kernel.

Findings

The new method improves accuracy in GLE simulations.

Several averages are exact with the proposed method.

Numerical experiments demonstrate superior performance over existing methods.

Abstract

We study numerical methods for the generalized Langevin equation (GLE) with a positive Prony series memory kernel, in which case the GLE can be written in an extended variable Markovian formalism. We propose a new splitting method that is easy to implement and is able to substantially improve the accuracy and robustness of GLE simulations in a wide range of the parameters. An error analysis is performed in the case of a one-dimensional harmonic oscillator, revealing that several averages are exact for the newly proposed method. Various numerical experiments in both equilibrium and nonequilibrium simulations are also conducted to compare the method with popular alternatives in interacting multi-particle systems.