Bifurcations of finite difference schemes and their approximate inertial forms

Rolf Bronstering; Min Chen

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (1998)

  • Volume: 32, Issue: 6, page 715-728
  • ISSN: 0764-583X

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Bronstering, Rolf, and Chen, Min. "Bifurcations of finite difference schemes and their approximate inertial forms." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 32.6 (1998): 715-728. <http://eudml.org/doc/193894>.

@article{Bronstering1998,
author = {Bronstering, Rolf, Chen, Min},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {approximate inertial form; Kuramota-Savashinsky equation; finite differences; multigrid algorithm; inertial manifolds; stability; bifurcations; convergence; reaction-diffusion equation},
language = {eng},
number = {6},
pages = {715-728},
publisher = {Dunod},
title = {Bifurcations of finite difference schemes and their approximate inertial forms},
url = {http://eudml.org/doc/193894},
volume = {32},
year = {1998},
}

TY - JOUR
AU - Bronstering, Rolf
AU - Chen, Min
TI - Bifurcations of finite difference schemes and their approximate inertial forms
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 1998
PB - Dunod
VL - 32
IS - 6
SP - 715
EP - 728
LA - eng
KW - approximate inertial form; Kuramota-Savashinsky equation; finite differences; multigrid algorithm; inertial manifolds; stability; bifurcations; convergence; reaction-diffusion equation
UR - http://eudml.org/doc/193894
ER -

References

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  1. [1] N. AUBRY, W. LIAN and E. S. TITI, Preserving symmetries in the proper orthogonal decomposition, SIAM J. Sci. Comp., 14, 483-505, 1993. Zbl0774.65084MR1204243
  2. [2] R. BRONSTERING, Some computational aspects on approximate inertial manifolds and finite differences. Discrete and Continuous Dynamical Systems, 2, 417-454, 1996. Zbl0949.65135MR1414079
  3. [3] M. CHEN, H. CHOI, T. DUBOIS, J. SHEN and R. TEMAM, The incremental unknowns-multilevel scheme for the simulation of turbulent channel flows. Proceedings of 1996 Summer Program, Center for Turbulence Research, NASA Ames/Stanford Univ., pages 291-308, 1996. 
  4. [4] M. CHEN and R. TEMAM. Incremental unknowns for solving partial differential equations. Numerische Mathematik, 59, 255-271, 1991. Zbl0712.65103MR1106383
  5. [5] M. CHEN and R. TEMAM, Nonlinear Galerkin method in the finite difference case and wavelet-like incremental-unknowns. Numerische Mathematik, 64(3), 271-294, 1993. Zbl0798.65093MR1206665
  6. [6] M. CHEN and R. TEMAM, Nonlinear Galerkin method with multilevel incremental-unknowns. In E.P. Agarwal, editor, Contributions in Numerical Mathematics, pages 151-164. WSSIAA, 1993. Zbl0834.65094MR1299757
  7. [7] E. J. DOEDEL, X. J. WANG and T. F. FAIRGRIEVE. Software for continuation and bifurcation problems in ordinary differential equations. CRPC-95-2, Center for Research on Parallel Computing, California Institute of Technology, 1995. 
  8. [8] C. FOIAS, JOLLY, KEVREKIDIS and E. S. TITI. Dissipativity of numerical schemes. Nonlinearity, pages 591-613, 1991. Zbl0734.65080MR1124326
  9. [9] C. FOIAS, O. MANLEY and R. TEMAM. Modeling of the interaction of small and large eddies in two dimensional turbulent flows. Math. Model. and Num. Anal., 22(1), 1988. Zbl0663.76054MR934703
  10. [10] C. FOIAS and E. S. TITIDetermining nodes, finite difference schemes and inertial manifolds. Nonlinearity, 4, 135-153, 1991. Zbl0714.34078MR1092888
  11. [11] G. GOLUB and C. VAN LOAN. Matrix Computations. The John Hopkins University Press, second edition, 1989. Zbl0733.65016MR1002570
  12. [12] J. HALE. Asymptotic Behavior of Dissipative Systems. AMS, 1988. Zbl0642.58013MR941371
  13. [13] J. M. HYMAN, B. NICOLAENKO and S. ZALESKIOrder and complexity in the Kuramoto-Sivashinsky model of weakly turbulent interfaces. Physica D, 23, 265-292, 1986. Zbl0621.76065MR876914
  14. [14] M. JOLLYExplicit construction of an inertial manifold for a reaction diffusion equation. J. Diff. Eq., 78, 220-261, 1989. Zbl0691.35049MR992147
  15. [15] M. S. JOLLY, I. G. KEVREKIDIS and E. S. TITI. Approximate inertial manifolds for the Kuramoto-Sivashinsky equation : Analysis and computations. Physica D, 44, 38-60, 1990. Zbl0704.58030MR1069671
  16. [16] M. S. JOLLY, I. G. KEVREKIDIS and E. S. TITI. Preserving dissipation in approximate inertial forms for the Kuramoto-Sivashinsky equation. J. Dyn. Diff. Eq., 3, 179-197, 1991. Zbl0738.35024MR1109435
  17. [17] D. A. JONES, L. G. MARGOLIN and E. S. TITI. On the effectiveness of the approximate inertial manifold-a computational study, to appear in Theoretical and Computational Fluid Dynamics, 1995. Zbl0838.76066
  18. [18] D. A. JONES, L. G. MARGOLIN and A. C. POJE. Enslaved finite difference schemes for nonlinear dissipative pdes. Num. Meth. for PDEs, page to appear. Zbl0879.65063MR1363860
  19. [19] KEVREKIDIS, NICOLAENKO and SCOVEL. Back in the saddle again : A computer assisted study of the Kuramoto-Sivashinsky equation. Siam J. Apl. Math., 50, 760-790, 1990. Zbl0722.35011MR1050912
  20. [20] E. KORONTINIS and M. R. TRUMMER. A finite difference scheme for Computing inertial manifolds. Z angew Math. Phys., 46, 419-444, 1995. Zbl0824.65125MR1335911
  21. [21] L. G. MARGOLIN and D. A. JONES. An approximate inertial manifold for computing Burger's equation. Physica D, 60, 175-184, 1992. Zbl0789.65069MR1195598
  22. [22] M. MARION, Approximate inertial manifolds for reaction-diffusion equations in high space dimension. J. Dyn. Diff. Eq., 1, 245-267, 1989. Zbl0702.35127MR1010967
  23. [23] M. MARION and R. TEMAM. Nonlinear Galerkin methods. SIAM J. Num. An., 26, 1139-1157, 1989. Zbl0683.65083MR1014878
  24. [24] B. NICOLAENKO, B. SCHEURER and R. TEMAM. Some global dynamical properties of the Kuramoto-Sivashinsky equation : Nonlinear stability and attractors. Physica D, 16, 155-183, 1985. Zbl0592.35013MR796268
  25. [25] R. TEMAM. Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer Verlag, 1988. Zbl0662.35001MR953967
  26. [26] R. WALLACE and D. M. SLOAN. Numerical solution of a nonlinear dissipative System using a pseudospectral method and inertial manifolds. Siam J. Sci. Comput., 16, 1049-1070, 1994. Zbl0833.65087MR1346292

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