Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-12-03T15:04:15.440Z Has data issue: false hasContentIssue false

Stability Analysis and Order Improvement for Time Domain Differential Quadrature Method

Published online by Cambridge University Press:  21 December 2015

Fangzong Wang*
Affiliation:
College of Electrical Engineering & New Energy, China Three Gorges University, Yichang 443002, China
Xiaobing Liao
Affiliation:
College of Electrical Engineering & New Energy, China Three Gorges University, Yichang 443002, China
Xiong Xie
Affiliation:
College of Electrical Engineering & New Energy, China Three Gorges University, Yichang 443002, China
*
*Corresponding author. Email:fzwang@ctgu.edu.cn (F. Z. Wang)
Get access

Abstract

The differential quadrature method has been widely used in scientific and engineering computation. However, for the basic characteristics of time domain differential quadrature method, such as numerical stability and calculation accuracy or order, it is still lack of systematic analysis conclusions. In this paper, according to the principle of differential quadrature method, it has been derived and proved that the weighting coefficients matrix of differential quadrature method meets the important V-transformation feature. Through the equivalence of the differential quadrature method and the implicit Runge-Kutta method, it has been proved that the differential quadrature method is A-stable and s-stage s-order method. On this basis, in order to further improve the accuracy of the time domain differential quadrature method, a class of improved differential quadrature method of s-stage 2s-order has been proposed by using undetermined coefficients method and Padé approximations. The numerical results show that the proposed differential quadrature method is more precise than the traditional differential quadrature method.

Type
Research Article
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bellman, R. and Casti, J., Differential quadrature and long term integration, J. Math. Anal. Appl., 34(2) (1971), pp. 235238.Google Scholar
[2]Bellman, R., Kashef, B. G. and Casti, J., Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations, J. Comput. Phys., 10 (1972), pp. 4052.Google Scholar
[3]Malik, M. and Civan, F., Comparative study of differential quadrature and cubature methods visàvis some conventional techniques in context of convection-diffusion-reaction problems, Chem. Eng. Sci., 50(3) (1995), pp. 531547.Google Scholar
[4]Bert, C. W. and Malik, M., Differential quadrature method in computational mechanics: a review, Appl. Mech. Rev., 49(1) (1996), pp. 128.Google Scholar
[5]Civan, F. and Sliepcevich, C. M., Application of differential quadrature to transport process, J. Math. Anal. Appl., 93 (1983), pp. 206221.CrossRefGoogle Scholar
[6]Jang, S. K., Bert, C. W. and Striz, A. G., Application of differential quadrature to static analysis of structural components, Int. J. Numer. Methods Eng., 28(3) (1989), pp. 561577.Google Scholar
[7]Bert, C. W., Wang, X. and Striz, A. G., Differential quadrature for static and free vibration analyses of anisotropic plates, Int. J. Solids Structures, 30(13) (1993), pp. 17371744.Google Scholar
[8]Du, H., Lim, M. K. and Lin, R. M., Application of generalized differential quadrature method to structural problems, Int. J. Numer. Methods Eng., 37(11) (1994), pp. 18811896.Google Scholar
[9]Xu, Q. W., Li, Z. F. and Wang, J., Modeling of transmission lines by the differential quadrature method, IEEE Microwave Guided Wave Letters, 9(4) (1999), pp. 145147.Google Scholar
[10]Xu, Q. W., Equivalent-circuit interconnects modeling based on the fifth-order differential quadrature methods, IEEE Transactions Very Large Scale Integration (VLSI) Systems, 11(3) (2003), pp. 10681079.Google Scholar
[11]Chen, W., Differential Quadrature Method and Its Applications in Engineering-Applying Special Matrix Product in Nonlinear Computation (in English), PhD thesis, Shanghai Jiao Tong University, 1997.Google Scholar
[12]Tanaka, M. and Chen, W., Coupling dual reciprocity BEM and differential quadrature method for time-dependent diffusion problems, Appl. Math. Model., 25(3) (2001), pp. 257268.Google Scholar
[13]Tanaka, M. and Chen, W., Dual reciprocity BEM applied to transient elastodynamic problems with differential quadrature method in time, Computer Methods Appl. Mech. Eng., 190(18-19) (2001), pp. 23312347.Google Scholar
[14]Fung, T. C., Solving initial value problems by differential quadrature method, Part 1: first order equations, Int. J. Numer. Methods Eng., 50(6) (2001), pp. 14111427.Google Scholar
[15]Fung, T. C., Solving initial value problems by differential quadrature method, Part 2: second and higher order equations, Int. J. Numer. Methods Eng., 50(6) (2001), pp. 14291454.3.0.CO;2-A>CrossRefGoogle Scholar
[16]Fung, T. C., Stability and accuracy of differential quadrature method in solving dynamic problems, Computer Methods Appl. Mech. Eng., 191 (2002), pp. 13111331.CrossRefGoogle Scholar
[17]Fung, T. C., On the equivalence of the time domain differential quadrature method and the dissipative Runge-Kutta collocation method, Int. J. Numer. Methods Eng., 53(2) (2002), pp. 409431.Google Scholar
[18]Hairer, E., Nørsett, S. P. and Wanner, G., Solving Ordinary Differential Equations I: Second Revised Edition, Springer, Berlin, 1992.Google Scholar
[19]Hairer, E. and Wanner, G., Solving Ordinary Differential Equations II: Second Revised Edition, Springer, Berlin, 1996.Google Scholar
[20]Butcher, J. C., Numerical Methods for Ordinary Differential Equations, Second Edition, Wiley, New York, 2008.Google Scholar
[21]Shu, C., Differential Quadrature and Its Application in Engineering, Springer, Berlin, 2000.CrossRefGoogle Scholar
[22]Wang, F. Z., Numerical Method for Transient Stability Computation of Large-Scale Power System, Science Press, Beijing, 2013.Google Scholar
[23]Wang, W. Q. and Li, S. F., The necessary and sufficient conditions of A-acceptability of high-order rational approximations to the function exp(z), Natural Science Journal of Xiangtan University, 22(1) (2000), pp. 47.Google Scholar
[24]Feng, K., Difference schemes for Hamiltonian formalism and symplectic geometry, J. Comput. Math., 4(3) (1986), pp. 279289.Google Scholar
[25]Feng, K. and Qin, M. Z., Hamiltonian algorithms for Hamiltonian dynamical systems, Progress Natural Sci., 1(2) (1991), pp. 105116.Google Scholar
[26]Feng, K. and Qin, M. Z., Symplectic Geometric Algorithm for Hamiltonian Systems, Zhejiang Science & Technology Press, Hangzhou, 2003.Google Scholar