Variational integrators of mixed order for dynamical systems with multiple time scales and split potentials - Wenger, Theresa; Ober-Blöbaum, Sina; Leyendecker, Sigrid (Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering, 2016)
The simulation of mechanical systems that act on multiple time scales, caused e.g. by different types or stiffnesses in potentials, is challenging as a stable integration of the fast dynamics requires a highly accurate approximation whereas for the simulation of the slow part a coarser approximation is accurate enough. With regard to the general goals of any numerical method, high accuracy and low computational costs, the presented variational integrators of mixed order couple coarse and fine approximations. The idea builds up on the higher order Galerkin variational integrators in [9] that are derived via Hamilton’s variational principle with a polynomial to approximate the configuration and an appropriate quadrature formula for the approximation of the integral of the Lagrangian. For the variational integration of systems with dynamics on multiple time scales, we use polynomials of different degrees to approximate the components that act on different time scales. Furthermore, quadrature formulas of different order approximate the integrals of the single energy contributions of the Lagrangian. This approach provides great flexibility in the design of the integrators. Their performance is investigated numerically by means of the Fermi-Pasta-Ulam problem and a numerical analysis regarding accuracy versus efficiency is carried out, where we focus on the integrators most promising to resolve the mentioned trade-of
Keywords: multiple time scales; higher order variational integrators; Fermi-Pasta-Ulam problem