Energy-preserving iteration schemes for Gauss collocation integrators

In this work, we develop energy-preserving iterative schemes for the (non-)linear systems arising in the Gauss integration of Poisson systems with quadratic Hamiltonian. Exploiting the relation between Gauss collocation integrators and diagonal Padé approximations, we establish a Krylov-subspace iteration scheme based on a Q -Arnoldi process for linear systems that provides energy conservation not only at convergence —as standard iteration schemes do—, but also at the level of the individual iterates. It is competitive with GMRES in terms of accuracy and cost for a single iteration step and hence offers significant efficiency gains, when it comes to time integration of high-dimensional Poisson systems within given error tolerances. On top of the linear results, we consider non-linear Poisson systems and design non-linear solvers for the implicit midpoint rule (Gauss integrator of second order), using the fact that the associated Padé approximation is a Cayley transformation. For the non-linear systems arising at each time step, we propose fixed-point and Newton-type iteration schemes that inherit the convergence order with comparable cost from their classical versions, but have energy-preserving iterates. • Development of energy-preserving Krylov-subspace based iterative schemes for linear systems arising in Gauss integration of linear Poisson systems. • Establishment of energy-preserving fixed-point and Newton-type iteration schemes for non-linear systems arising in the implicit numerical integration of non-linear Poisson systems with quadratic Hamiltonian via midpoint rule.