A general framework for symplectic geometric integration for stochastically excited Hamiltonian systems on manifolds

This work introduces anew integration scheme in the field of Stochastic Geometric Integration on Manifolds, dedicated to precise modeling of real-world dynamic phenomena characterized by inherent randomness. Conventional numerical techniques often encounter difficulties when confronted with uncertainty and the preservation of manifold geometry, consequently leading to inaccuracies. In response to these challenges, the method proposed considers geometric precision, stochastic dynamics, and symplecticity all-together. Drawing inspiration from methods based on Lie groups, this work formulates a framework that can adapt to stochastic systems, aligning itself with the principles of geometric stochastic differential equations. By incorporating higher-order geometric integration schemes possessing inherent symplectic properties, the study achieves accurate solutions that simultaneously retain intricate system geometry, maintain symplecticity, and accommodate inherent randomness. The method maintains its general flavor by virtue of a completely intrinsic formulation which can be adopted to broad class of manifolds rather than a specific one. Through numerous representative benchmarks, the study substantiates the effectiveness of the proposed scheme. The results demonstrate a significant enhancement in the numerical scheme's performance, owing to its utilization of the inherent symplectic nature.