Chebyshev spectral variational integrator and applications  

作　　者：Zhonggui YI Baozeng YUE Mingle DENG 

机构地区：[1]School of Aerospace Engineering,Beijing Institute of Technology,Beijing 100081,China [2]China Academy of Space Technology,Beijing 100094,China

出　　处：《Applied Mathematics and Mechanics(English Edition)》2020年第5期753-768,共16页应用数学和力学（英文版）

基　　金：the National Natural Science Foundation of China (Nos. 11472041,11532002,11772049,and 11802320)。

摘　　要：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.

关 键 词：geometric numerical method spectral method variational integrator Clenshaw-Curtis quadrature rule barycentric Lagrange interpolation orbital propagation 

分 类 号：O302[理学—力学] P138.2[天文地球—天体力学]