In this paper,Chebyshev interpolation nodes and barycentric Lagrange interpolation basis function are used to deduce the scheme for solving the Helmholtz equation.First of all,the interpolation basis function is appli...In this paper,Chebyshev interpolation nodes and barycentric Lagrange interpolation basis function are used to deduce the scheme for solving the Helmholtz equation.First of all,the interpolation basis function is applied to treat the spatial variables and their partial derivatives,and the collocation method for solving the second order differential equations is established.Secondly,the differential matrix is used to simplify the given differential equations on a given test node.Finally,based on three kinds of test nodes,numerical experiments show that the present scheme can not only calculate the high wave numbers problems,but also calculate the variable wave numbers problems.In addition,the algorithm has the advantages of high calculation accuracy,good numerical stability and less time consuming.展开更多
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 rat...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.展开更多
基金partially supported by National Natural Science Foundation of China(11772165,11961054,11902170)Key Research and Development Program of Ningxia(2018BEE03007)+1 种基金National Natural Science Foundation of Ningxia(2018AAC02003,2020AAC03059)Major Innovation Projects for Building First-class Universities in China’s Western Region(Grant No.ZKZD2017009).
文摘In this paper,Chebyshev interpolation nodes and barycentric Lagrange interpolation basis function are used to deduce the scheme for solving the Helmholtz equation.First of all,the interpolation basis function is applied to treat the spatial variables and their partial derivatives,and the collocation method for solving the second order differential equations is established.Secondly,the differential matrix is used to simplify the given differential equations on a given test node.Finally,based on three kinds of test nodes,numerical experiments show that the present scheme can not only calculate the high wave numbers problems,but also calculate the variable wave numbers problems.In addition,the algorithm has the advantages of high calculation accuracy,good numerical stability and less time consuming.
基金the National Natural Science Foundation of China (Nos. 11472041,11532002,11772049,and 11802320)。
文摘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.
