Abstract Accurate time integrators that preserving Birkhoffian structure are of great practical use for Birkhoffian systems. In this paper, a class of structure-preserving discontinuous Galerkin variational integrators (DGVIs) is presented. Start from the Pfaff action functional, the technique of variational integrators combined with discontinuous Galerkin time discretization is used to derive numerical schemes for Birkhoffian systems. For the derived DGVIs, symplecticity is proved rigorously through the preserving of particular 2-forms induced by these integrators. Linear stability and order of accuracy of the DGVIs are illustrated considering the example of linear damped oscillators. The order of accuracy and the property of preserving conserved quantities of the developed DGVIs are also confirmed by numerical examples. Comparisons are made with several numerical schemes such as backward/forward Euler, Runge–Kutta and RBF methods to show the advantages of DGVIs in preserving the Birkhoffians.

中文翻译：

---

Birkhoff系统的一类保结构不连续Galerkin变分时间积分器

摘要 保留 Birkhoffian 结构的精确时间积分器对于 Birkhoffian 系统具有重要的实际用途。在本文中，提出了一类保持结构的不连续伽辽金变分积分器（DGVI）。从 Pfaff 作用泛函开始，使用变分积分器技术结合不连续 Galerkin 时间离散化来推导出 Birkhoffian 系统的数值方案。对于导出的 DGVI，辛性通过保留这些积分器诱导的特定 2-形式而得到严格证明。以线性阻尼振荡器为例，说明了 DGVI 的线性稳定性和精度顺序。数值例子也证实了所开发的 DGVI 的精度顺序和保存守恒量的特性。