A parallel iterative method for variational integration

Discrete variational methods show excellent performance in numerical simulations of different mechanical systems. In this paper, we introduce an iterative procedure for the solution of discrete variational equations for boundary value problems. It consists in repeatedly correcting the position of the non-boundary points of a proposed discrete path in order to converge to a solution. This is accomplished using a parallelization strategy that leverages the capabilities of multicore CPUs and GPUs. Further, we develop this parallel method for higher-order Lagrangian systems, which appear in fully-actuated problems and beyond. Convergence conditions for this kind of method are investigated. We illustrate their excellent behavior in some interesting examples, namely Zermelo's navigation problem, a fuel-optimal navigation problem, interpolation problems, or in a fuel optimization problem for a controlled 4-body problem in astrodynamics, showing the potential of our method.