High order conformal symplectic and ergodic schemes for stochastic Langevin equation via generating functions

TL;DR

This paper develops high-order conformal symplectic numerical schemes for stochastic Langevin equations with additive noise, ensuring geometric structure preservation and ergodic properties, verified through theoretical proofs and numerical experiments.

Contribution

It introduces a novel methodology to construct high weak order conformal symplectic schemes by modifying generating functions, applicable to stochastic Langevin equations.

Findings

The proposed scheme dissipates symplectic form exponentially.

For linear systems, the scheme inherits ergodicity.

Numerical results confirm theoretical properties.

Abstract

In this paper, we consider the stochastic Langevin equation with additive noises, which possesses both conformal symplectic geometric structure and ergodicity. We propose a methodology of constructing high weak order conformal symplectic schemes by converting the equation into an equivalent autonomous stochastic Hamiltonian system and modifying the associated generating function. To illustrate this approach, we construct a specific second order numerical scheme, and prove that its symplectic form dissipates exponentially. Moreover, for the linear case, the proposed scheme is also shown to inherit the ergodicity of the original system, and the temporal average of the numerical solution is a proper approximation of the ergodic limit over long time. Numerical experiments are given to verify these theoretical results.