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

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.