Fast, Variational Newmark Integrators on Manifolds

Nawaf Bou-Rabee, California Institute of Technology

In this talk a novel technique to design robust integrators for long-time simulations of Hamiltonian systems with a Lie Group symmetry is presented. From this technique a fast, variational Newmark integrator is derived. Due to the integrator’s variational construction, it is discrete symplectic two-form preserving. As a consequence the discrete energy remains bounded for long-time integrations. Moreover, the method is explicit and hence computationally efficient.

Numerical results in the form of Poincaré sections and work-precision diagrams for integrable and nonintegrable examples are provided to support the theory. The nonintegrable examples demonstrate that standard projection methods (methods which project onto the momentum and energy level-sets) do not capture the right dynamical behavior for long-time integrations in comparison to the proposed methods. Additional comparisons to the current state-of-the-art reveal that this method is fast and efficient.

Abstract Author(s): Nawaf Bou-Rabee