Solving Langevin equation with the bicolour rooted tree method

TL;DR

This paper introduces a systematic high-order algorithm for solving Langevin equations using the bicolour rooted tree method, validated through test problems involving energy relaxation and Ornstein-Uhlenbeck noise.

Contribution

It develops a novel high-order numerical scheme for Langevin equations based on stochastic Taylor expansion and the bicolour rooted tree method.

Findings

The algorithm accurately models energy relaxation in double well potentials.

It effectively handles time-dependent Langevin equations with Ornstein-Uhlenbeck noise.

Compared to existing methods, it offers improved accuracy and versatility.

Abstract

Stochastic differential equations, especially the one called Langevin equation, play an important role in many fields of modern science. In this paper, we use the bicolour rooted tree method, which is based on the stochastic Taylor expansion, to get the systematic pattern of the high order algorithm for Langevin equation. We propose a popular test problem, which is related to the energy relaxation in the double well, to test the validity of our algorithm and compare our algorithm with other usually used algorithms in simulations. And we also consider the time-dependent Langevin equation with the Ornstein-Uhlenbeck noise as our second example to demonstrate the versatility of our method.