Summary
This work presents a geometry preserving weak symplectic stochastic integrator for long-time simulation of Hamiltonian systems on S^2 manifold. The proposed algorithm preserves the energy of the system accurately for long-time integration of the system using a multiple time stepping strategy. For the geometric systems under stochastic perturbation, the proposed symplectic scheme is formulated using an Ito-Taylor expansion-based weak integration scheme and exploiting the relation between the manifold and its Lie Algebra through exponential mapping. Since the weak schemes are concerned with accuracy with respect to a function of the true solution rather than pathwise accuracy, the weak schemes have the advantage of considerably simpler and cheaper numerical implementation. Numerical illustrations on non-linear oscillators on the S^2 manifold are demonstrated to show the geometric and symplectic nature of the algorithm.
