Abstract
The Chebyshev spectral variational integrator (CSVI) is presented in this paper. Spectral methods have aroused great interest in approximating numerically a smooth problem for their attractive geometric convergence rates. The geometric numerical methods are praised for their excellent long-time geometric structure-preserving properties. According to the generalized Galerkin framework, we combine two methods together to construct a variational integrator, which captures the merits of both methods. Since the interpolating points of the variational integrator are chosen as the Chebyshev points, the integration of Lagrangian can be approximated by the Clenshaw-Curtis quadrature rule, and the barycentric Lagrange interpolation is presented to substitute for the classic Lagrange interpolation in the approximation of configuration variables and the corresponding derivatives. The numerical float errors of the first-order spectral differentiation matrix can be alleviated by using a trigonometric identity especially when the number of Chebyshev points is large. Furthermore, the spectral variational integrator (SVI) constructed by the Gauss-Legendre quadrature rule and the multi-interval spectral method are carried out to compare with the CSVI, and the interesting kink phenomena for the Clenshaw-Curtis quadrature rule are discovered. The numerical results reveal that the CSVI has an advantage on the computing time over the whole progress and a higher accuracy than the SVI before the kink position. The effectiveness of the proposed method is demonstrated and verified perfectly through the numerical simulations for several classical mechanics examples and the orbital propagation for the planet systems and the Solar system.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
FENG, K. and QIN, M. Z. Symplectic Geometric Algorithms for Hamiltonian Systems, Springer, Berlin (2010)
MARSDEN, J. E. and WEST, M. Discrete mechanics and variational integrators. Acta Numerica, 10, 357–514 (2001)
LEOK, M. and SHINGEL, T. General techniques for constructing variational integrators. Frontiers of Mathematics in China, 7(2), 273–303 (2012)
OBER-BLÖBAUM, S. and SAAKE, N. Construction and analysis of higher order Galerkin variational integrators. Advances in Computational Mathematics, 41(6), 955–986 (2015)
LEOK, M. Generalized Galerkin variational integrators. arXiv, 0508360 (2005) https://arxiv.org/abs/math/0508360
LEE, T., LEOK, M., and MCCLAMROCH, N. H. Lie group variational integrators for the full body problem in orbital mechanics. Celestial Mechanics and Dynamical Astronomy, 98(2), 121–144 (2007)
PALACIOS, L. and GURFIL, P. Variational and symplectic integrators for satellite relative orbit propagation including drag. Celestial Mechanics and Dynamical Astronomy, 130(4), 31 (2018)
OBER-BLÖBAUM, S., JUNGE, O., and MARSDEN, J. E. Discrete mechanics and optimal control: an analysis. ESAIM: Control, Optimisation and Calculus of Variations, 17(2), 322–352 (2011)
KOBILAROV, M. B. and MARSDEN, J. E. Discrete geometric optimal control on Lie groups. IEEE Transactions on Robotics, 27(4), 641–655 (2011)
MOORE, A., OBER-BLÖBAUM, S., and MARSDEN, J. E. Trajectory design combining invariant manifolds with discrete mechanics and optimal control. Journal of Guidance, Control, and Dynamics, 35(5), 1507–1525 (2012)
BOLATTI, D. A. and DE RUITER, A. H. Galerkin variational integrators for orbit propagation with applications to small bodies. Journal of Guidance, Control, and Dynamics, 42(2), 347–363 (2018)
HALL, J. and LEOK, M. Lie group spectral variational integrators. Foundations of Computational Mathematics, 17(1), 199–257 (2017)
HE, L., WU, H. B., and MEI, F. X. Variational integrators for fractional Birkhoffian systems. Nonlinear Dynamics, 87(4), 2325–2334 (2017)
TREFETHEN, L. N. Spectral Methods in MATLAB, Society of Industrial and Applied Mathematics, Philadelphia (2000)
BOYD, J. P. Chebyshev and Fourier Spectral Methods, 2nd ed., Dover Publications, Inc., New York (2001)
SHEN, J., TANG, T., and WANG, L. L. Spectral Methods: Algorithms, Analysis and Applications, Springer-Verlag, Berlin (2011)
HALE, N. and TREFETHEN, L. N. Chebfun and numerical quadrature. Science China Mathematics, 55(9), 1749–1760 (2012)
DRISCOLL, T. A., HALE, N., and TREFETHEN, L. N. Chebfun Guide, Pafnuty Publications, Oxford (2014)
JIAO, Y. J. and GUO, B. Y. Mixed spectral method for exterior problems of Navier-Stokes equations by using generalized Laguerre functions. Applied Mathematics and Mechanics (English Edition), 30(5), 561–574 (2009) https://doi.org/10.1007/s10483-009-0503-z
LI, B. and CHEN, S. Direct spectral domain decomposition method for 2D incompressible Navier-Stokes equations. Applied Mathematics and Mechanics (English Edition), 36(8), 1073–1090 (2015) https://doi.org/10.1007/s10483-015-1964-7
GONG, Q., ROSS, I. M., and FAHROO, F. Costate computation by a Chebyshev pseudospectral method. Journal of Guidance, Control, and Dynamics, 33(2), 623–628 (2010)
GE, X. S., YI, Z. G., and CHEN, L. Q. Optimal control of attitude for coupled-rigid-body spacecraft via Chebyshev-Gauss pseudospectral method. Applied Mathematics and Mechanics (English Edition), 38(9), 1257–1272 (2017) https://doi.org/10.1007/s10483-017-2236-8
YI, Z. G. and GE, X. S. Attitude maneuver of dual rigid bodies spacecraft using ftp-adaptive pseudo-spectral method. International Journal of Aeronautical and Space Sciences, 20(1), 214–227 (2019)
HALL, J. and LEOK, M. Spectral variational integrators. Numerische Mathematik, 130(4), 681–740 (2015)
TREFETHEN, L. N. Is Gauss quadrature better than Clenshaw-Curtis? SIAM Review, 50(1), 67–87 (2008)
BERRUT, J. P. and TREFETHEN, L. N. Barycentric Lagrange interpolation. SIAM Review, 46(3), 501–517 (2004)
LI, Y. Q., WU, B. Y., and LEOK, M. Construction and comparison of multidimensional spectral variational integrators and spectral collocation methods. Applied Numerical Mathematics, 132, 35–50 (2018)
LI, Y. Q., WU, B. Y., and LEOK, M. Spectral-collocation variational integrators. Journal of Computational Physics, 332, 83–98 (2017)
LIU, W. J., WU, B. Y., and SUN, J. Some numerical algorithms for solving the highly oscillatory second-order initial value problems. Journal of Computational Physics, 276, 235–251 (2014)
WEIDEMAN, J. A. C. and TREFETHEN, L. N. The kink phenomenon in Fejér and Clenshaw-Curtis quadrature. Numerische Mathematik, 107(4), 707–727 (2007)
SOMMARIVA, A. Fast construction of Fejér and Clenshaw-Curtis rules for general weight functions. Computers & Mathematics with Applications, 65(4), 682–693 (2013)
WALDVOGEL, J. Fast construction of the Fejér and Clenshaw-Curtis quadrature rules. BIT Numerical Mathematics, 46(1), 195–202 (2006)
Author information
Authors and Affiliations
Corresponding author
Additional information
Project supported by the National Natural Science Foundation of China (Nos. 11472041, 11532002, 11772049, and 11802320)
Rights and permissions
About this article
Cite this article
Yi, Z., Yue, B. & Deng, M. Chebyshev spectral variational integrator and applications. Appl. Math. Mech.-Engl. Ed. 41, 753–768 (2020). https://doi.org/10.1007/s10483-020-2602-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10483-020-2602-8
Key words
- geometric numerical method
- spectral method
- variational integrator
- Clenshaw-Curtis quadrature rule
- barycentric Lagrange interpolation
- orbital propagation