A new symplectic integrator for stochastic Hamiltonian systems on manifolds

This paper presents a significant development in the field of geometric numerical stochastic integration schemes. Specifically, the Geometric Symplectic Ito-Taylor 1.0 strong numerical integration scheme, tailored for Hamiltonian systems evolving on 2-sphere or S2 manifold is proposed. The core of this proposed algorithm centers around a new weak symplectic condition. This condition ensures the numerical stability for long duration time integration that can be readily achieved through numerical steps rather than relying solely on analytical satisfaction of the condition. This proposed advancement over the state-of-the-art caters to improved accuracy and stability of numerical simulations in manifold-based dynamical systems for long duration simulations. Through extensive analysis and numerical experiments, the effectiveness and reliability of the proposed scheme and symplectic criterion are validated. The findings of this study promises to offer valuable insights for researchers and practitioners working in the field of symplectic numerical integration methods.