High order conformal symplectic and ergodic schemes for stochastic Langevin equation via generating functions
For researchers in stochastic geometric numerical integration, this provides a new method to preserve both conformal symplecticity and ergodicity in high-order schemes.
The paper develops high-order conformal symplectic schemes for the stochastic Langevin equation with additive noise, constructing a second-order scheme that preserves exponential dissipation of the symplectic form and inherits ergodicity for linear systems, with numerical verification.
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.