Computational coarse graining of a randomly forced one-dimensional Burgers equation

Physics of Fluids

Volumn 20, Issue 3, 2008, Pages

  Ahuja, Sunil ^a   Yakhot, Victor ^b   Kevrekidis, Ioannis G ^a  

^a Princeton University   (United States)

^b BOSTON UNIVERSITY   (United States)

Author keywords

Computational fluid dynamics; Differential equations; Turbulence

Indexed keywords

COMPUTER SIMULATION; DIFFERENTIAL EQUATIONS; NUMERICAL METHODS; SPECTRUM ANALYSIS; TURBULENCE;

COARSE GRAINING; ENERGY SPECTRUM; LARGE-SCALE FEATURES;

COMPUTATIONAL FLUID DYNAMICS;

COMPUTATIONAL FLUID DYNAMICS; COMPUTER SIMULATION; DIFFERENTIAL EQUATIONS; NUMERICAL METHODS; SPECTRUM ANALYSIS; TURBULENCE;

EID: 41949133110     PISSN: 10706631     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.2856212     Document Type: Article

References (36)

1. I. G. Kevrekidis, C. W. Gear, J. M. Hyman, P. G. Kevrekidis, O. Runborg, and C. Theodoropoulos, "Equation-free coarse-grained multiscale computation: Enabling microscopic simulators to perform system-level tasks," Commun. Math. Sci. ZZZZZZ1539-6746 1, 715 (2003).
2. C. Theodoropoulos, Y.-H. Qian, and I. G. Kevrekidis, "'Coarse' stability and bifurcation analysis using time steppers: A reaction-diffusion example," Proc. Natl. Acad. Sci. U.S.A. PNASA60027-842410.1073/pnas.97.18.9840 97, 9840 (2000).
3. I. G. Kevrekidis, C. W. Gear, and G. Hummer, "Equation-free: The computer-assisted analysis of complex, multiscale systems," AIChE J. AICEAC0001-154110.1002/aic.10106 50, 1346 (2004).
4. A. Chekhlov and Y. Yakhot, "Kolmogorov turbulence in a random-force-driven Burgers equation," Phys. Rev. E PLEEE81063-651X10.1103/PhysRevE.51.R2739 51, R2739 (1995).
5. D. Forster, D. Nelson, and M. Stephen, "Large-distance and long-time properties of a randomly stirred fluid," Phys. Rev. A PLRAAN1050-294710.1103/PhysRevA.16.732 16, 732 (1977).
6. U. Frisch, Turbulence (Cambridge University Press, Cambridge, 1996).
7. Y. Yakhot and A. Chekhlov, "Algebraic tails of probability density functions in the random-force-driven Burgers turbulence," Phys. Rev. Lett. PRLTAO0031-900710.1103/PhysRevLett.77.3118 77, 3118 (1996).
8. A. Polyakov, "Turbulence without pressure," Phys. Rev. E PLEEE81063-651X10.1103/PhysRevE.52.6183 52, 6183 (1995).
9. W. E, K. Khanin, A. Mazel, and Y. Sinai, "Probability distribution functions for the random forced Burgers equation," Phys. Rev. Lett. PRLTAO0031-900710.1103/PhysRevLett.78.1904 78, 1904 (1997).
10. W. E and E. Vanden Eijnden, "Asymptotic theory for the probability density functions in Burgers turbulence," Phys. Rev. Lett. PRLTAO0031-900710.1103/PhysRevLett.83.2572 83, 2572 (1999).
11. J. Bec and U. Frisch, "Probability distribution functions of derivatives and increments for decaying Burgers turbulence," Phys. Rev. E PLEEE81063-651X10.1103/PhysRevE.61.1395 61, 1395 (2000).
12. A. Chekhlov and Y. Yakhot, "Kolmogorov turbulence in a random-force-driven Burgers equation: Anomalous scaling and probability density functions," Phys. Rev. E PLEEE81063-651X10.1103/ PhysRevE.52.5681 52, 5681 (1995).
13. D. Mitra, J. Bec, R. Pandit, and U. Frisch, "Is multiscaling an artifact in the stochastically forced Burgers equation?" Phys. Rev. Lett. PRLTAO0031-900710.1103/PhysRevLett.94.194501 94, 194501 (2005).
14. Y. Yakhot and Z.-S. She, "Long-time, large-scale properties of the random-force-driven Burgers equation," Phys. Rev. Lett. PRLTAO0031-900710.1103/PhysRevLett.60.1840 60, 1840 (1988).
15. M. Kardar, G. Parisi, and Y.-C. Zhang, "Dynamic scaling of growing interfaces," Phys. Rev. Lett. PRLTAO0031-900710.1103/ PhysRevLett.58.882 58, 882 (1986).
16. D. A. Kessler, I. G. Kevrekidis, and L. Chen, "Equation-free dynamic renormalization of a KPZ-type equation," Phys. Rev. E PLEEE81063-651X10.1103/PhysRevE.73.036703 73, 036703 (2006).
17. Y. Yakhot and S. A. Orszag, "Renormalization group analysis of turbulence," J. Sci. Comput. JSCOEB0885-747410.1007/BF01061452 1, 3 (1986).
18. H. W. Wyld, "Formulation of the theory of turbulence in an incompressible fluid," Phys. Rev. PHRVAO0031-899X10.1103/ PhysRev.14.143 14, 143 (1961).
19. C. W. Gear and I. G. Kevrekidis, "Projective methods for stiff differential equations: Problems with gaps in their eigenvalue spectrum," SIAM J. Sci. Comput. (USA) SJOCE31064-827510.1137/ S1064827501388157 24, 1091 (2003).
20. C. W. Gear and I. G. Kevrekidis, "Telescopic projective integrators for stiff differential equations," J. Comput. Phys. JCTPAH0021-999110.1016/S0021-9991(03)00082-2 187, 95 (2003).
21. C. W. Gear, I. G. Kevrekidis, and C. Theodoropoulos, "'Coarse' integration/bifurcation analysis via microscopic simulators: Micro-Galerkin methods," Comput. Chem. Eng. CCENDW0098-135410.1016/ S0098-1354(02)00020-0 26, 941 (2002).
22. A. J. Chorin, O. H. Hald, and R. Kupferman, "Optimal prediction and the Mori-Zwanzig representation of irreversible processes," Proc. Natl. Acad. Sci. U.S.A. PNASA60027-842410.1073/pnas.97.7.2968 97, 2968 (2000).
23. M. E. Kavousanakis, R. Erban, A. G. Boudouvis, C. W. Gear, and I. G. Kevrekidis, "Projective and coarse projective integration for problems with continuous symmetries," J. Comput. Phys. JCTPAH0021-999110.1016/j.jcp.2006.12.003 225, 382 (2007).
24. L. Chen, P. G. Debenedetti, C. W. Gear, and I. G. Kevrekidis, "From molecular dynamics to coarse self-similar solutions: A simple example using equation-free computation," J. Non-Newtonian Fluid Mech. JNFMDI0377-025710.1016/j.jnnfm.2003.12.007 120, 215 (2004).
25. Y. Zou, I. G. Kevrekidis, and R. Ghanem, "Equation-free dynamic renormalization: Self-similarity in multidimensional particle system dynamics," Phys. Rev. E PLEEE81063-651X10.1103/PhysRevE.72.046702 72, 046702 (2005).
26. L. Chen, I. G. Kevrekidis, and P. G. Kevrekidis, "Equation-free dynamic renormalization in a glassy compaction model," Phys. Rev. E PLEEE81063-651X10.1103/PhysRevE.74.016702 74, 016702 (2006).
27. D. G. Aronson, S. I. Betelu, and I. G. Kevrekidis, "Going with the flow: A Lagrangian approach to self-similar dynamics and its consequences," http://arxiv.org/abs/nlin.AO/0111055 (2001).
28. C. W. Rowley, I. G. Kevrekidis, J. E. Marsden, and K. Lust, "Reduction and reconstruction for self-similar dynamical systems," Nonlinearity NONLE50951-771510.1088/0951-7715/16/4/304 16, 1257 (2003).
29. C. T. Kelley, I. G. Kevrekidis, and L. Qiao, "Newton-Krylov solvers for timesteppers," http://arxiv.org/math/0404374 (2004).
30. S. L. Lee and C. W. Gear, "On-the-fly local error estimation for projective integrators," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. SJNAAM0036-1429 (submitted).
31. S. L. Lee and C. W. Gear, "Second-order accurate projective integrators for multiscale problems," J. Comput. Appl. Math. JCAMDI0377-042710.1016/j.cam.2006.02.018 201, 258 (2007).
32. R. R. Coifman, S. Lafon, A. B. Lee, M. Maggioni, B. Nadler, F. Warner, and S. Zucker, "Geometric diffusion as a tool for harmonic analysis and structure definition of data: Diffusion maps," Proc. Natl. Acad. Sci. U.S.A. PNASA60027-842410.1073/pnas.0500334102 102, 7426 (2005).
33. R. R. Coifman, S. Lafon, A. B. Lee, M. Maggioni, B. Nadler, F. Warner, and S. Zucker, "Geometric diffusion as a tool for harmonic analysis and structure definition of data: Multiscale methods," Proc. Natl. Acad. Sci. U.S.A. PNASA60027-842410.1073/pnas.0500896102 102, 7432 (2005).
34. B. Nadler, S. Lafon, R. C. Coifman, and I. G. Kevrekidis, "Diffusion maps, spectral clustering and the reaction coordinates of dynamical systems," Appl. Comput. Harmon. Anal. ACOHE91063-520310.1016/j.acha.2005.07.004 21, 113 (2006).
35. C. Rosales and C. Meneveau, "A minimal multiscale Lagrangian map approach to synthesize non-Gaussian turbulent vector fields," Phys. Fluids PHFLE61070-663110.1063/1.2227003 18, 075104 (2006).
36. J. Li, P. G. Kevrekidis, C. W. Gear, and I. G. Kevrekidis, "Deciding the nature of the coarse equation through microscopic simulation: The baby-bathwater scheme," Multiscale Model. Simul. MMSUBT1540-345910.1137/S1540345902413322 1, 391 (2003).