Hostname: page-component-848d4c4894-p2v8j Total loading time: 0.001 Render date: 2024-06-03T12:21:57.597Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence of Linear Multistep Methods and One-Leg Methods for Index-2 Differential-Algebraic Equations with a Variable Delay

Published online by Cambridge University Press:  03 June 2015

Hongliang Liu  and
Aiguo Xiao
Hongliang Liu*
Affiliation:
School of Mathematics and Computational Science, Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education, Xiangtan University, Xiangtan, Hunan 411105, P. R. China
Aiguo Xiao*
Affiliation:
School of Mathematics and Computational Science, Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education, Xiangtan University, Xiangtan, Hunan 411105, P. R. China
*
Email: lhl@xtu.edu.cn
Corresponding author. Email: xag@xtu.edu.cn
Get access

Abstract

Linear multistep methods and one-leg methods are applied to a class of index-2 nonlinear differential-algebraic equations with a variable delay. The corresponding convergence results are obtained and successfully confirmed by some numerical examples. The results obtained in this work extend the corresponding ones in literature.

Keywords

65L8065L20index-2 differential-algebraic equationsvariable delaylinear mutistep methodsone-leg methodsconvergence
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 4 , Issue 5 , October 2012 , pp. 636 - 646
Copyright
Copyright © Global-Science Press 2012

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]Arévalo, C., Führer, C. and Söderlind, G., Stabilized multistep methods for index 2 Euler-Lagrange differential-algebraic equations, BIT, 36 (1996), pp. 1–13.CrossRefGoogle Scholar
[2]Brenan, B. K. E. and Engquist, B. E., Backward differentiation approximations of nonlinear differential-algebraic equations and supplement, Math. Comput., 51 (1988), pp. 659–676.CrossRefGoogle Scholar
[3]Xiao, A. and Liu, J., Error analysis of parallel multivalue hybrid methods for index-2 differential-algebraic equations, Int. J. Comput. Sci. Math., 2 (2008), pp. 181–199.CrossRefGoogle Scholar
[4]Arévalo, C., Führer, C. and Söderlind, G., β-blocked multistep methods for Euler-Lagrange DAEs: linear analysis, Z. Angew. Math. Mech., 77 (1997), pp. 609–617.CrossRefGoogle Scholar
[5]Arévalo, C., Führer, C. and Söderlind, G., β-blocking of difference corrected multistep methods for nonstiff index-2 DAEs, Appl. Numer. Math., 35 (2000), pp. 293–305.CrossRefGoogle Scholar
[6]Cao, Y. and Li, Q., Highest order multistep formula for solving index 2 differential-algebraic equations, BIT, 38 (1998), pp. 644–662.CrossRefGoogle Scholar
[7]Hairer, E. and Wanner, G., Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Equations problems, 2nd Edition, Springer, Berlin, 1996.CrossRefGoogle Scholar
[8]Xu, Y. and Zhao, J., Stability of Runge-Kutta methods for neutral delay-integro-differential-algebraic system, Math. Comput. Simul., 79 (2008), pp. 571–583.CrossRefGoogle Scholar
[9]Li, H. and Li, J., Blocks implicit one-step methods for a class of retarded differential algebraic equation, J. Huazhong Univ. Sci. Tech., 31 (2003), pp. 111–113 (in Chinese).Google Scholar
[10]Xiao, F. and Zhang, C., Convergence analysis of one-leg methods for a class retarded differential algebraic equation, J. Numer. Meth. Comput. Appl., 29 (2008), pp. 217–225 (in Chinese).Google Scholar
[11]Zhu, W. and Pezold, L. R., Asymptotic stability of Hessenberg delay differential-algebraic equations of retarded or neutral type, Appl. Numer. Math., 27 (1998), pp. 309–325.CrossRefGoogle Scholar
[12]Ascher, U. M. and Pezold, L. R., The numerical solution of delay-differential-algebraic equations of retarded and neutral type, SIAM J. Mumer. Anal., 32 (1995), pp. 1635–1657.CrossRefGoogle Scholar
[13]Hauber, R., Numerical treatment of retarded differential-algebraic equations by collocation methods, Adv. Comput. Math., 7 (1997), pp. 573–592.CrossRefGoogle Scholar
[14]Liu, H. and Xiao, A., Convergence of backword differentiation formulas for index-2 differential-algebraic equations with variable delay, Chin. J. Eng. Math., 28 (2011), pp. 335–342 (in Chinese).Google Scholar
[15]Hairer, E.Nørsett, S. P., and Wanner, G., Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd Edition, Springer, Berlin, 1993.Google Scholar