A parallel iterative method for variational integration

Sebastián J. Ferraro; David Martín de Diego; Rodrigo T. Sato Martín de Almagro

doi:10.3934/jcd.2025006

Article Contents

2025, Volume 12, Issue 4: 528-553. Doi: 10.3934/jcd.2025006

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A parallel iterative method for variational integration

1.
Instituto de Matemática (INMABB), Departamento de Matemática, Universidad Nacional del Sur (UNS) – CONICET, Bahía Blanca, Argentina
2.
Instituto de Ciencias Matemáticas, ICMAT (CSIC-UAM-UC3M-UCM), Madrid, Spain
3.
Institute of Applied Dynamics, Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, Germany

^*Corresponding author: Sebastián J. Ferraro
^*Corresponding author: Sebastián J. Ferraro

Received: December 2024

Revised: March 2025

Early access: July 2025

Published online: July 2025

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 70-08, 70G45, 70Hxx; Secondary: 70Q05, 49-XX, 49Mxx, 49M20, 65K10, 65L10, 65L20.

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Figure 1. An iteration of the parallel method for $ N = 3 $

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Figure 2. Several local solutions to the optimal time navigation problem starting from $ (0,0) $ and ending at $ (6,2) $. The time for each trajectory is shown

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Figure 3. A minimal fuel trajectory for a fixed total duration $ T = 30 $, joining $ (0,0) $ to $ (6,5) $, with $ N = 200 $

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Figure 4. An optimal trajectory for the second-order problem with interpolation nodes

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Figure 5. An optimal trajectory with varying wind. The black dot is the ship, and the black vectors along the trajectory represent the wind that it will encounter in its journey. Each figure shows the vector field $ W $ at different times

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Figure 6. Spacecraft trajectory starting from a geosynchronous orbit (in black, before starting the planned trajectory) and parking at the $ \mathrm{L}_5 $ Lagrange point of the Earth-Moon system (to scale). The trajectory is displayed in a rotating frame with the Earth and the Moon fixed in the diagram. Distances are in astronomical units (au). On the left plot, the instantaneous fuel consumption is shown using grayscale

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Figure 7. Left: adaptive step size. Right: fixed step size

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