Patch dynamics with buffers for homogenization problems

Journal of Computational Physics

Volumn 213, Issue 1, 2006, Pages 264-287

  Samaey, Giovanni ^a   Kevrekidis, Ioannis G ^b   Roose, Dirk ^a  

^a KU Leuven   (Belgium)

^b Princeton University   (United States)

Author keywords

Equation free methods; Gap tooth scheme; Homogenization; Multi scale computation; Patch dynamics

Indexed keywords

DIFFUSION IN LIQUIDS; DYNAMICS; HOMOGENIZATION METHOD; LINEAR EQUATIONS;

EQUATION-FREE METHODS; GAP-TOOTH SCHEME; HOMOGENIZATION; HOMOGENIZATION PROBLEMS; MULTI-SCALE COMPUTATION; MULTI-SCALES; PATCH DYNAMICS; SPACE AND TIME; SPACE SCALE; TIME-SCALES;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 31744447017     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2005.08.010     Document Type: Article

References (43)

1. A. Abdulle W. E Finite difference heterogeneous multi-scale method for homogenization problems J. Comput. Phys. 191 1 2003 18-39
2. G. Allaire Homogenization and two-scale convergence SIAM J. Math. Anal. 23 6 1992 1482-1518
3. I. Babuska Homogenization and its applications B. Hubbard 1975 SYNSPADE 89-116
4. M. Bär, Räumliche Strukturbildung bei einer Oberflächenreaktion. Chemische Wellen und Turbulenz in der CO-Oxidation auf Platin-Einkristal-oberflächen, Ph.D. Thesis, Freie Universität Berlin, 1993.
5. M. Bär A. Bangia I. Kevrekidis G. Haas H.-H. Rotermund G. Ertl Composite catalyst surfaces: Effect of inert and active heterogeneities on pattern formation J. Phys. Chem. 100 1996 19106-19117
6. A. Bensoussan J. Lions G. Papanicolaou Asymptotic Analysis of Periodic Structures: Studies in Mathematics and its Applications vol. 5 1978 North Holland Amsterdam
7. D. Cioranescu P. Donato An Introduction to Homogenization 1999 Oxford University Press Oxford
8. M. Dorobantu B. Engquist Wavelet-based numerical homogenization SIAM J. Numer. Anal. 35 2 1998 540-559
9. W. E B. Engquist The heterogeneous multi-scale methods Comm. Math. Sci. 1 1 2003 87-132
10. J. Elezgaray G. Berkooz P. Holmes Large-scale statistics of the Kuramoto-Sivashinsky equation: A wavelet-based approach Phys. Rev. E 54 1996 224-230
11. B. Engquist Computation of oscillatory solutions to hyperbolic differential equations Springer Lecture Notes Math. 1270 1987 10-22
12. B. Engquist O. Runborg Wavelet-based numerical homogenization with applications Multiscale and Multiresolution Methods: Lecture Notes in Computational Science and Engineering vol. 20 2002 Springer Berlin 97-148
13. C. Gear I. Kevrekidis Projective methods for stiff differential equations: Problems with gaps in their eigenvalue spectrum SIAM J. Sci. Comput. 24 4 2003 1091-1106 http://www.neci.nj.nec.com/homepages/cwg/ projective.pdf
14. C.W. Gear, I.G. Kevrekidis, A legacy-code approach to low-dimensional computation: Constraint-defined manifolds, J. Sci. Comp. 25(1) (2006), in press.
15. C. Gear J. Li I. Kevrekidis The gap-tooth method in particle simulations Phys. Lett. A 316 2003 190-195 arxiv:cOnd-mat/0211455
16. M. Graham I. Kevrekidis K. Asakura J. Lauterbach K. Krischer H.-H. Rotermund G. Ertl Effects of boundaries on pattern formation: Catalytic oxidation of CO on Platinum Science 264 1994 80-82
17. N. Hadjiconstantinou Hybrid atomistic-continuum formulations and the moving contact-line problem J. Comput. Phys. 154 1999 245-265
18. A.C. Hindmarsh ODEPACK, A Systematized Collection of ODE Solvers R.S. Stepleman Scientific Computing 1983 North-Holland Amsterdam 55-64
19. T. Hou X. Wu A multiscale finite element method for elliptic problems in composite materials and porous media J. Comput. Phys. 134 1997 169-189
20. T. Hou X. Wu Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients Math. Comput. 68 227 1999 913-943
21. G. Hummer I. Kevrekidis Coarse molecular dynamics of a peptide fragment: free energy, kinetics and long-time dynamics computations J. Chem. Phys. 118 23 2003 10762-10773 arxiv:pHysics/0212108
22. E. Jones, T. Oliphant, P. Peterson, et al., SciPy: Open source scientific tools for Python (2001). Available from: .
23. J. Keener Homogenization and propagation in the bistable equation Physica D 136 2000 1-17
24. I. Kevrekidis, Coarse bifurcation studies of alternative microscopic/ hybrid simulators, Plenary lecture, CAST Division, AIChE Annual Meeting, Los Angeles, 2000, slides can be obtained from: .
25. I. Kevrekidis C. Gear J. Hyman P. Kevrekidis O. Runborg C. Theodoropoulos Equation-free multiscale computation: Enabling microscopic simulators to perform system-level tasks Comm. Math. Sci. 1 4 2003 715-762
26. I. Kevrekidis B. Nicolaenko J. Scovel Back in the saddle again: A computer assisted study of the Kuramoto-Sivashinsky equation SIAM J. Appl. Math. 50 1990 760-790
27. J. Li P. Kevrekidis C. Gear I. Kevrekidis Deciding the nature of the "coarse equation" through microscopic simulations: The baby-bathwater scheme SIAM Multiscale Model. Simul. 1 3 2003 391-407
28. J. Li D. Liao S. Yip Imposing field boundary conditions in MD simulation of fluids: Optimal particle controller and buffer zone feedback Mat. Res. Soc. Symp. Proc. 538 1998 473-478
29. A. Makeev D. Maroudas I. Kevrekidis Coarse stability and bifurcation analysis using stochastic simulators: Kinetic Monte Carlo examples J. Chem. Phys. 116 2002 10083-10091
30. A. Makeev D. Maroudas A. Panagiotopoulos I. Kevrekidis Coarse bifurcation analysis of kinetic Monte Carlo simulations: A lattice-gas model with lateral interactions J. Chem. Phys. 117 18 2002 8229-8240
31. A. Matache I. Babuska C. Schwab Generalized p-FEM in homogenization Numerische Mathematik 86 2 2000 319-375
32. N. Neuss W. Jäger G. Wittum Homogenization and multigrid Computing 66 1 2001 1-26
33. O. Runborg C. Theodoropoulos I. Kevrekidis Effective bifurcation analysis: A time-stepper based approach Nonlinearity 15 2002 491-511
34. G. Samaey I. Kevrekidis D. Roose Damping factors for the gap-tooth scheme S. Attinger P. Koumoutsakos Multiscale Modelling and Simulation Lecture Notes in Computational Science and Engineering vol. 36 2004 Springer Berlin 93-102
35. G. Samaey D. Roose I. Kevrekidis The gap-tooth scheme for homogenization problems SIAM Multiscale Model. Simul. 4 1 2005 278-306
36. C. Schwab A. Matache Multiscale and Multiresolution Methods: Lecture Notes in Computational Science and Engineering vol. 20 2002 Springer-Verlag New York 197-238
37. S. Setayeshar C. Gear H. Othmer I. Kevrekidis Application of coarse integration to bacterial chemotaxis SIAM Multiscale Model. Simul. 4 1 2005 307-327
38. C. Siettos A. Armaou A. Makeev I. Kevrekidis Microscopic/stochastic timesteppers and coarse control: A kinetic Monte Carlo example AIChE J. 49 7 2003 1922-1926 arxiv:nLin.CG/0207017
39. C. Siettos M. Graham I. Kevrekidis Coarse Brownian dynamics for nematic liquid crystals: Bifurcation, projective integration and control via stochastic simulation J. Chem. Phys. 118 22 2003 10149-10157 arxiv:cOnd-mat/0211455
40. S. Shvartsman E. Schütz R. Imbihl I. Kevrekidis Dynamics on microcomposite catalytic surfaces: The effect of active boundaries Phys. Rev. Lett. 83 1999 2857-2860
41. C. Theodoropoulos, Y. Qian, I. Kevrekidis, Coarse stability and bifurcation analysis using time-steppers: A reaction-diffusion example, in: Proceedings of the National Academy of Sciences USA, vol. 97, 2000.
42. C. Theodoropoulos K. Sankaranarayanan S. Sundaresan I. Kevrekidis Coarse bifurcation studies of bubble flow Lattice-Boltzmann simulations Chem. Eng. Sci. 59 2003 2357-2362 arxiv:nLin.PS/0111040
43. P. Van Leemput K. Lust I.G. Kevrekidis Coarse-grained numerical bifurcation analysis of lattice Boltzmann models Physica D: Nonlinear Phenomena 210 1-2 2005 58-76