Accurately model the Kuramoto - Sivashinsky dynamics with holistic discretization

SIAM Journal on Applied Dynamical Systems

Volumn 5, Issue 3, 2006, Pages 365-402

  MacKenzi, T ^a   Roberts, A J ^a  

^a UNIVERSITY OF SOUTHERN QUEENSLAND   (Australia)

Author keywords

Computational discretizations; Kuramoto sivashinsky equation; Low dimensional modeling; Multi scale modeling

Indexed keywords

DYNAMICAL SYSTEMS; DYNAMICS; SCHRODINGER EQUATION; SPUTTERING;

CENTER MANIFOLD THEORIES; CENTER MANIFOLDS; COARSE GRIDS; COMPUTATIONAL DISCRETIZATIONS; DISCRETIZATION; DISCRETIZATIONS; DYNAMICAL EQUATIONS; INTERNAL BOUNDARIES; KURAMOTO-SIVASHINSKY EQUATION; LOW-DIMENSIONAL MODELING; MULTI-SCALE MODELING; NUMERICAL SOLUTIONS; PHYSICAL DOMAINS; SIVASHINSKY EQUATIONS; SPATIO TEMPORALS; STEADY STATES;

NONLINEAR EQUATIONS;

EID: 57249110723     PISSN: None     EISSN: 15360040     Source Type: Journal    
DOI: 10.1137/050627733     Document Type: Article

References (25)

1. C. FOIAS, M. S. JOLLY, I. G. KEVREKIDIS. G. R. SELL, AND E. S. TITI, On the computation of inertial manifolds, Phys Lett. A, 131 (1988), pp. 433-436.
2. C. FOIAS, M. S. JOLLY, I. G. KEVREKIDIS, AND E. S. TITI, On some dissipative fully discrete nonlinear Galerkin schemes for the Kuramoto-Sivashinsky equation, Phys. Lett. A, 186 (1994), pp. 87-96.
3. C. FOIAS, B. NLCOLAENKO, G. R. SELL, AND R. TEMAN, Inertial manifolds for the Kuramoto-Sivashinsky equation, C. R. Acad. Sci. Paris Sér. I Math., 301 (1985), pp. 285-288 (in French).
4. P. HOLMES, J. L. LUMLEY, AND G. BERKOOZ, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, Cambridge, UK, 1996.
5. J. M. HYMAN AND B. NICOLAENKO, The Kuramoto-Sivashinsky equation: A bridge between PDEs and dynamical systems, Phys. D, 18 (1986), pp. 113-126.
6. M. S. JOLLY, I. G. KEVREKIDIS, AND E. S. TLTI, Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: Analysis and computations, Phys. D, 44 (1990), pp. 38-60.
7. I. G. KEVREKIDIS, B. NICOLAENKO, AND J. C. SCOVEL, Back in the saddle again: A computer assisted study of the Kuramoto-Sivashinsky equation, SIAM J. Appl. Math., 50 (1990), pp. 760-790.
8. Y. KURAMOTO, Diffusion induced chaos in reactions systems, Progr. Theoret. Phys. Suppl., 64 (1978), pp. 346-367.
9. S. J. LIAO, The quite general BEM for strongly nonlinear problems, in Boundary Elements XII, C. A. Brebbia, S. Kim, T. A. Osswald, and H. Power, eds., Computational Mechanics Publications, Southampton, MA, 1995, pp. 67-74.
10. T. MACKENZIE AND A. J. ROBERTS, Holistic finite differences accurately model the dynamics of the Kuramoto-Sivashinsky equation, ANZIAM J., 42 (2000), pp. C918-C935.
11. T. MACKENZIE, Create Accurate Numerical Models of Complex Spatio-temporal Dynamical Systems with Holistic Discretisation, Ph.D. thesis, University of Southern Queensland, Toowoomba, Australia, 2005.
12. M. MARION AND R. TEMAM, Nonlinear Galerkin methods, SIAM J. Numer. Anal., 26 (1989), pp. 1139-1157.
13. Y. POMEAU AND S. ZALESKI, The Kuramoto-Sivashinsky equation: A caricature of hydradynamic turbulence?, in Macroscopic Modelling of Turbulent Flows, Lecture Notes in Phys. 230, Springer, Berlin, 1985, pp. 296-303.
14. A. J. ROBERTS, The utility of an invariant manifold description of the evolution of a dynamical system, SIAM J. Math. Anal., 20 (1989), pp. 1447-1458.
15. A. J. ROBERTS, Low-Dimensional Modelling of Dynamical Systems, technical report, 1997, available online at http://arXiv.org/abs/ chao-dyn/9705010.
16. A. J. ROBERTS, Low-dimensional modelling of dynamics via computer algebra, Comput. Phys. Comm., 100 (1997), pp. 215-230.
17. A. J. ROBERTS, Holistic discretisation ensures fidelity to Burgers' equation, Appl. Numer. Math., 37 (2001), pp. 371-396.
18. A. J. ROBERTS, A holistic finite difference approach models linear dynamics consistently, Math. Comp., 72 (2002), pp. 247-262.
19. A. J. ROBERTS, Derive boundary conditions for holistic discretisations of Burgers' equation, in Proc. of 10th Computational Techniques and Applications Conference CTAC-2001, K. Burrage and R. B. Sidje, eds., ANZIAM J., 44 (2003), pp. C664-C686.
20. A. J. ROBERTS, Holistic Discretisation of Dynamical Partial Differential Equations, technical report, version 1.1, 2003, available online at http://www.sci.usq.edu.au/staff/aroberts/holistic.htinl.
21. A. J. ROBERTS AND I. G. KEVREKIDIS, Higher order accuracy in the gap-tooth scheme for large-scale dynamics using microscopic simulators, in Proc. of 12th Computational Techniques and Applications Conference CTAC-2004, R. May and A. J. Roberts, eds., ANZIAM J., 46 (2005), pp. C637-C657.
22. J. C. ROBINSON, The asymptotic completeness of inertial manifolds, Nonlinearity, 9 (1996), pp. 1325-1340.
23. C. SCOVEL, I. G. KEVREKIDIS, AND B. NICOLAENKO, Scaling laws and the prediction of bifurcations in systems modeling pattern formation, Phys. Lett. A, 130 (1988), pp. 73-80.
24. G. I. SIVASHINSKY, Nonlinear analysis of hydrodynamic instability in laminar flames, Part I: Derivation of the basic equations, Acta Astronaut., 4 (1977), pp. 1176-1206.
25. A. VANDERBAUWHEDE, Centre manifolds, normal forms, and elementary bifurcation, in Dynamics Reported, Wiley, Chichester, UK, 1989, Vol. 2, pp. 89-169.