Symplectic methods for stochastic Hamiltonian systems: asymptotic error distributions and Hamiltonian-specific analysis

TL;DR

This paper analyzes the asymptotic error distributions of symplectic numerical methods for stochastic Hamiltonian systems, demonstrating their advantages for long-term simulations through Hamiltonian-specific error analysis.

Contribution

It introduces a new approach to compute asymptotic error distributions and characterizes their relation to Hamiltonian properties, highlighting the superiority of symplectic methods.

Findings

Asymptotic error distributions satisfy Hamiltonian equations.

New method links stochastic modified equations to error distributions.

Symplectic methods outperform non-symplectic ones in long-term Hamiltonian simulations.

Abstract

In this paper, we investigate the asymptotic error distributions of symplectic methods for stochastic Hamiltonian systems and further provide Hamiltonian-specific analysis that clarifies the superiority of symplectic methods. Our contribution is threefold. First, we derive the asymptotic error distributions of symplectic methods for stochastic Hamiltonian systems with multiplicative noise and additive noise, respectively, and show that the obtained limiting stochastic processes satisfy equations retaining the Hamiltonian formulations. Second, we propose a new approach for calculating the asymptotic error distribution, revealing the connection between the stochastic modified equation and the asymptotic error distribution. Third, we characterize the limiting distribution of the normalized Hamiltonian deviation, thereby illustrating through test equations the superiority of symplectic methods for long-time simulations of the Hamiltonians, even in the limit as the step size tends to zero.