Title: Approximating Symplectic Realizations: A General Framework for the Construction of Poisson Integrators Show Chinese title

Title: 近似辛实现：构造泊松积分器的一般框架

Abstract: While the construction of symplectic integrators for Hamiltonian dynamics is well understood, an analogous general theory for Poisson integrators is still lacking. The main challenge lies in overcoming the singular and non-linear geometric behavior of Poisson structures, such as the presence of symplectic leaves with varying dimensions. In this paper, we propose a general approach for the construction of geometric integrators on any Poisson manifold based on independent geometric and dynamic sources of approximation. The novel geometric approximation is obtained by adapting structural results about symplectic realizations of general Poisson manifolds. We also provide an error analysis for the resulting methods and illustrative applications. Show Chinese abstract

Abstract: 虽然对哈密顿动力学的辛积分器的构造已经得到了很好的理解，但关于泊松积分器的类似一般理论仍然缺乏。 主要挑战在于克服泊松结构的奇异和非线性几何行为，例如存在不同维数的辛叶层。 在本文中，我们提出了一种一般方法，基于独立的几何和动态近似源，在任何泊松流形上构造几何积分器。 新颖的几何近似是通过适应关于一般泊松流形的辛实现的结构结果得到的。 我们还为所得到的方法提供了误差分析和示例应用。