A class of structure-preserving discontinuous Galerkin variational time integrators for Birkhoffian systems

Highlights：

• Symplecticity of the DGVIs is proved rigorously.

• Linear stability of the DGVIs is illustrated considering the example of linear damped oscillators, error estimates are also given for the first-order, the second-order and the third-order DGVIs. The order of accuracy and preservation of Birkhoffians are numerically confirmed for all the three DGVIs.

• Comparisons are made against backward/forward Euler, Runge-Kutta and RBF methods to show the advantages of DGVIs in preserving the Birkhoffians.

摘要

•Combining variational formulation with discontinuous Galerkin time dis-cretization, a class of stable, structure preserving discontinuous Galerkin variational integrators (DGVIs) is proposed, which can be generalized to arbitrary high order to approximate the Birkhoffian systems.•Symplecticity of the DGVIs is proved rigorously.•Linear stability of the DGVIs is illustrated considering the example of linear damped oscillators, error estimates are also given for the first-order, the second-order and the third-order DGVIs. The order of accuracy and preservation of Birkhoffians are numerically confirmed for all the three DGVIs.•Comparisons are made against backward/forward Euler, Runge-Kutta and RBF methods to show the advantages of DGVIs in preserving the Birkhoffians.

论文评审过程：Received 20 April 2020, Revised 24 September 2020, Accepted 15 October 2020, Available online 3 November 2020, Version of Record 3 November 2020.