Non-standard finite-difference methods for vibro-impact problems

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 461, Issue 2058, 2005, Pages 1927-1950

  Dumont, Yves ^a   Lubuma, Jean M S ^a,b  

^a UNIVERSITÉ DE LA RÉUNION   (France)

^b UNIVERSITY OF PRETORIA   (South Africa)

Author keywords

Energy preserving schemes; Impact oscillators; Non smooth mechanics; Non standard finite difference method; Qualitative stability; Vibro impact scheme

Indexed keywords

EID: 22344458038     PISSN: 13645021     EISSN: 14712946     Source Type: Journal    
DOI: 10.1098/rspa.2004.1425     Document Type: Article

References (30)

1. Al-Kahby, H., Dannan, F. & Elaydi, S. 2000 Non-standard discretization methods for some biological models. In Applications of nonstandard finite difference schemes (ed. R. E. Mickens), pp. 155-180. Singapore: World Scientific.
2. Anguelov, R. & Lubuma, J. M.-S. 2001 Contributions to the mathematics of the nonstandard finite difference method and applications. Numer. Meth. Part. Differ. Equ. 17, 518-543.
3. Anguelov, R. & Lubuma, J. M.-S. 2003 Nonstandard finite difference methods by nonlocal approximation. Math. Comput. Simul. 61, 465-475.
4. Anguelov, R., Lubuma, J. M.-S. & Mahudu, S. K. 2003 Qualitatively stable finite difference schemes for advection-reaction equations. J. Comput. Appl. Math. 158, 19-30.
5. Anguelov, R., Kama, P. & Lubuma, J. M.-S. 2005 On non-standard finite difference models of reaction-diffusion equations. J. Comput. Appl. Math. 175, 11-29.
6. Brogliato, B. 2000 Nonsmooth mechanics: models, dynamics, 2nd edn. Control Communications and Control Engineering Series. London: Springer-Verlag.
7. Budd, C., Dux, F. & Cliff, A. 1995 The effect of frequency and clearance variations on single-degree-of-freedom impact oscillators. J. Sound Vib. 184, 475-502.
8. Dumont, Y. 2002 Vibrations of a beam between stops: simulations and comparison of several numerical schemes. Math. Comput. Simul. 60, 45-83.
9. Dumont, Y. 2003 Some remarks on a vibro-impact scheme. Numer. Algorithms 33, 227-240.
10. Gautschi, W. 1961 Numerical solution of ordinary differential equations based on trigonometric polynomials. Numer. Math. 3, 381-397.
11. Goeleven, D., Motreanu, D., Dumont, Y. & Rochdi, M. 2003 Variational and hemivariational inequalities: theory, methods and applications, Vol. I. Unilateral analysis and unilateral mechanics. Nonconvex Optimization and Its Applications Series, vol. 69. London: Kluwer Academic Publishers.
12. Gumel A. B. (Guest ed.) 2003 Special issue of the Journal of Difference Equations and Applications, Series vol. 9, dedicated to Prof. R. E. Mickens on the occasion of his 60th birthday.
13. Gumel, A. B., Mickens, R. E. & Corbett, B. D. 2003 A non-standard finite difference scheme for a model of HIV transmission and control. J. Comput. Meth. Sci. Eng. 3, 91-98.
14. Hairer, E., Lubich, C. & Wanner, G. 2002 Geometric numerical integration: structure-preserving algorithms for ordinary differential equations. Computational Mathematics Series. Berlin: Springer-Verlag.
15. Janin, O. & Lamarque, C. H. 2001 Comparison of several numerical methods for mechanical systems with impacts. Int. J. Numer. Meth. Eng. 51, 1101-1132.
16. Lambert, J. D. 1991 Numerical methods for ordinary differential systems. New York: John Wiley & Sons.
17. Lubuma, J. M.-S. & Roux, A. 2003 An improved theta method for systems of ordinary differential equations. J. Differ. Equ. Appl. 9, 1023-1035.
18. Mickens, R. E. 1994 Nonstandard finite difference models of difference equations. Singapore: World Scientific.
19. Mickens, R. E. 1997 Relation between the time and space step-sizes in nonstandard finite-difference schemes for the Fisher equation. Numer. Meth. Part. Differ. Equ. 13, 51-55.
20. Mickens, R. E. 1999 Nonstandard finite difference schemes: a status report. In Theoretical and computational acoustics 97 (ed. Y. C. Teng, E. C. Shang, Y. H. Pao, M. H. Schultz & A. D. Pierce), pp. 419-428. Singapore: World Scientific.
21. Mickens, R. E. (ed.) 2000 Applications of nonstandard finite difference schemes. Singapore: World Scientific.
22. Mickens, R. E. 2002 Nonstandard finite difference schemes for differential equations. J. Differ. Equ. Appl. 8, 823-847.
23. Moghadas, S. M., Alexander, M. E., Corbet, B. D. & Gumel, A. B. 2003 A positivity-preserving Mickens-type discretization of an epidemic model. J. Differ. Equ. Appl. 9, 1037-1051.
24. Paoli, L. 1993 Analyse numérique de vibrations avec contraintes unilatérales. Ph.D. thesis, Université Claude Bernard, Lyon, France.
25. Paoli, L. 2001 Time discretization of vibro-impact. Phil. Trans. R. Soc. A 359, 2405-2428.
26. Paoli, L. & Schatzman, M. 1993 Mouvement àun nombre fini de degrés de liberté avec contraintes unilatérales: Cas avec perte d'énergie. RAIRO, Modélisation Math. Anal. Numér. 27, 673-717.
27. Paoli, L. & Schatzman, M. 2002a A numerical scheme for impact problems. I. The one-dimensional case. SIAM J. Numer. Anal. 40, 702-733.
28. Paoli, L. & Schatzman, M. 2002b A numerical scheme for impact problems. II. The multidimensional case. SIAM J. Numer. Anal. 40, 734-768.
29. Peterka, F. 2000 Dynamics of double impact oscillators. Facta Universitatis 2, 1177-1190.
30. Stuart, A. M. & Humphries, A. R. 1998 Dynamical systems and numerical analysis. New York: Cambridge University Press.