Pseudo-conformal structure preserving explicit numerical methods for the stochastic linearly damped Hamiltonian systems

Abstract

In this paper, the linearly damped stochastic differential equations with invariants are examined. These invariants follow a linear differential equation with coefficients that are either linear constants or time-dependent. To preserve the essential characteristics of these linearly damped stochastic differential equations, a stochastic exponential integrator is utilized. Moreover, the stochastic pseudo-conformal symplectic methods are constructed and their pseudo-conformal symplectic orders for the stochastic damped Hamiltonian systems with additive noises are analyzed. All of these methods are explicit so that the implementations become more easier than implicit methods. Particularly, these methods have desired properties in accuracy and approximately preserved symplectic structure of the systems through some numerical experiments, especially including Schrödinger equation.