High-order symplectic Lie group methods on $ SO(n) $ using the polar decomposition

Xuefeng Shen; Khoa Tran; Melvin Leok

doi:10.3934/jcd.2022003

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2022, Volume 9, Issue 4: 529-551. Doi: 10.3934/jcd.2022003

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High-order symplectic Lie group methods on $ SO(n) $ using the polar decomposition

Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0112, USA

^* Corresponding author: Melvin Leok
^* Corresponding author: Melvin Leok

Received: June 30, 2018

Revised: December 31, 2021

Early access: February 2022

Published: October 2022

This research has been supported in part by NSF under grants DMS-1411792, DMS-1345013, DMS-1813635, CCF-2112665, by AFOSR under grant FA9550-18-1-0288, and by the DoD under grant HQ00342010023 (Newton Award for Transformative Ideas during the COVID-19 Pandemic)

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Abstract

A variational integrator of arbitrarily high-order on the special orthogonal group $ SO(n) $ is constructed using the polar decomposition and the constrained Galerkin method. It has the advantage of avoiding the second order derivative of the exponential map that arises in traditional Lie group variational methods. In addition, a reduced Lie–Poisson integrator is constructed and the resulting algorithms can naturally be implemented by fixed-point iteration. The proposed methods are validated by numerical simulations on $ SO(3) $ which demonstrate that they are comparable to variational Runge–Kutta–Munthe-Kaas methods in terms of computational efficiency. However, the methods we have proposed preserve the Lie group structure much more accurately and and exhibit better near energy preservation.

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Mathematics Subject Classification: Primary: 37M15, 65P10, 70H65, 70H25.

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Figure 1. Order comparison plots between VRKMK and VPD methods: The black-dashed lines are references for the corresponding orders

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Figure 2. Long-term energy error for second order VRKMK/VPD methods

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Figure 3. Long-term energy error for third order VRKMK/VPD methods

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Figure 4. Long-term energy error for fourth order VRKMK/VPD methods

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Figure 5. Long-term energy error for sixth order VRKMK/VPD methods

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Figure 6. Orthogonality error for sixth order VRKMK/VPD methods

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Figure 7. Run-time comparison for fourth and sixth order VRKMK/VPD methods

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Table 1. Butcher tableaux for the comparison tests

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