Equation-free, multiscale computation for unsteady random diffusion

Multiscale Modeling and Simulation

Volumn 4, Issue 3, 2005, Pages 915-935

  Xiu, Dongbin ^a,b   Kevrekidis, Ioannis G ^a  

^a Princeton University   (United States)

^b PURDUE UNIVERSITY   (United States)

Author keywords

Diffusion in random media; Equation free; Multiscale problem; Stochastic modeling

Indexed keywords

COMPUTATIONAL METHODS; COMPUTER SIMULATION; DIFFERENTIAL EQUATIONS; INTEGRATION; MATHEMATICAL MODELS; POLYNOMIALS; RANDOM PROCESSES;

EQUATION-FREE; MULTISCALE PROBLEM; STOCHASTIC MODELING;

DIFFUSION;

EID: 33748432259     PISSN: 15403459     EISSN: 15403467     Source Type: Journal    
DOI: 10.1137/040615006     Document Type: Article

References (42)

1. I. Babuška and P. Chatzipantelidis, On solving elliptic stochastic partial differential equations, Comput. Methods Appl. Mech. Engrg., 191 (2002), pp. 4093-4122.
2. I. Babuška, R. Tempone, and G. E. Zouraris, Galerkin finite element approximations of stochastic elliptic partial differential equations, SIAM J. Numer. Anal., 42 (2004), pp. 800-825.
3. G. P. A. Bensoussan and J.-L. Lions, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978.
4. D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford University Press, Oxford, UK, 1999.
5. G. Dagan, Flow and Transport in Porous Formations, Springer-Verlag, Heidelberg, Berlin, New York, 1989.
6. M. Deb, I. Babuška, and J. Oden, Solution of Stochastic partial differential equations using Galerkin finite element techniques, Comput. Methods Appl. Mech. Engrg., 190 (2001), pp. 6359-6372.
7. W. E, Homogenization of linear and nonlinear transport equations, Comm. Pure Appl. Math., 45 (1992), pp. 301-326.
8. G. Fishman, Monte Carlo: Concepts, Algorithms, and Applications, Springer-Verlag, New York, 1996.
9. R. Freeze, A stochastic-conceptual analysis of one-dimensional groundwater flow in nonuniform homogeneous media, Water Resour. Res., 11 (1975), pp. 725-741.
10. C. W. Gear, T. J. Kaper, I. G. Kevrekidis, and A. Zagaris, Projecting to a slow manifold: Singularly perturbed systems and legacy codes, SIAM J. Appl. Dyn. Syst., 4 (2005), pp. 711-732.
11. C. W. Gear and I. G. Kevrekidis, Projective methods for stiff differential equations: Problems with gaps in their eigenvalue spectrum, SIAM J. Sci. Comput., 24 (2003), pp. 1091-1106.
12. C. Gear and I. Kevrekidis, Telescopic projective methods for parabolic differential equations, J. Comput. Phys., 187 (2003), pp. 95-109.
13. C. Gear and I. Kevrekidis, Constraint-defined manifolds: A legacy-code approach to low- dimensional computations, J. Sci. Comput., submitted.
14. C. Gear, J. Li, and I. Kevrekidis, The gap-tooth method in particle simulations, Phys. Lett. A, 316 (2003), pp. 190-195.
15. R. Ghanem and P. Spanos, Stochastic Finite Elements: A Spectral Approach, Springer-Verlag, New York, 1991.
16. T. Hou and X.-H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys., 134 (1997), pp. 169-189.
17. T. Hou, X.-H. Wu, and Z. Cai, Convergence of a multi-scale finite element method for elliptic problems with rapidly oscillating coefficients, Math. Comp., 68 (1999), pp. 913-943.
18. T. Hughes, G. R. Feijoo, L. Mazzei, and J.-B. Quincy, The variational multiscale method - a paradigm for computational mechanics, Comput. Methods Appl. Mech. Engrg., 166 (1998), pp. 3-24.
19. G. Karniadakis and S. Sherwin, Spectral/hp Element Methods for CFD, Oxford University Press, New York, 1999.
20. I. Kevrekidis, C. Gear, and G. Hummer, Equation-free: The computer-assisted analysis of complex, multiscale systems, AIChE J., 50 (2004), pp. 1346-1354.
21. I. Kevrekidis, C. Gear, J. Hyman, P. Kevrekidis, O. Runborg, and C. Theodoropoulos, Equation-free coarse-grained multiscale computation: Enabling microscopic simulators to perform system-level analysis, Commun. Math. Sci., 1 (2003), pp. 715-762.
22. M. Kleiber and T. Hien, The Stochastic Finite Element Method, John Wiley & Sons, Chichester, UK, 1992.
23. O. Le Maitre, O. Knio, H. Najm, and R. Ghanem, Uncertainty propagation using Wiener-Haar expansions, J. Comput. Phys., 197 (2004), pp. 28-57.
24. O. Le Maitre, H. Najm, R. Ghanem, and O. Knio, Multi-resolution analysis of Wiener-type uncertainty propagation schemes, J. Comput. Phys., 197 (2004), pp. 502-531.
25. J. Li, P. G. Kevrekidis, C. W. Gear, and I. G. Kevrekidis, Deciding the nature of the coarse equation through microscopic simulations: The baby-bathwater scheme, Multiscale Model. Simul., 1 (2003), pp. 391-407.
26. W. Liu, G. Besterfield, and A. Mani, Probabilistic finite elements in nonlinear structural dynamics, Comput. Methods Appl. Mech. Engrg., 56 (1986), pp. 61-81.
27. M. Loève, Probability Theory, 4th ed., Springer-Verlag, New York, Heidelberg, 1977.
28. A. Makeev, D. Maroudas, and I. Kevrekidis, Coarse stability and bifurcation analysis using stochastic simulators: Kinetic Monte Carlo examples, J. Chem. Phys., 116 (2002), pp. 10083-10091.
29. A. Makeev, D. Maroudas, A. Panagiotopoulos, and I. Kevrekidis, Coarse bifurcation analysis of kinetic Monte Carlo simulations: A lattice-gas model with lateral interactions, J. Chem. Phys., 117 (2002), pp. 8229-8240.
30. A. Matache, I. Babuška, and C. Schwab, Generalized p-FEM in homogenization, Numer. Math., 86 (2000), pp. 319-375.
31. G. Milton, Theory of Composites, Cambridge University Press, Cambridge, UK, 2002.
32. A. Oberai and P. Pinsky, A multiscale finite element method for the Helmholtz equation, Comput. Methods Appl. Mech. Engrg., 154 (1998), pp. 281-297.
33. P. Renard and G. De Marsily, Calculating equivalent permeability: A review, Adv. Water Res., 20 (1997), pp. 253-278.
34. R. Rico-Martinez, C. Gear, and I. Kevrekidis, Coarse projective kMC integration: Forward/reverse initial and boundary value problems, J. Comput. Phys., 196 (2004), pp. 474-489.
35. O. Runborg, C. Theodoropoulos, and I. Kevrekidis, Effective bifurcation analysis: A time-stepper based approach, Nonlinearity, 15 (2002), pp. 491-511.
36. G. Samaey, D. Roose, and I. G. Kevrekidis, The gap-tooth scheme for homogenization problems, Multiscale Model. Simul., 4 (2005), pp. 278-306.
37. S. Sirisup, D. Xiu, G. Karniadakis, and I. Kevrekidis, Equation-free/ Galerkin-free POD- assisted computation of incompressible flows, J. Comput. Phys., 207 (2005), pp. 568-587.
38. C. Theodoropoulos, Y. Qian, and I. Kevrekidis, Coarse stability and bifurcation analysis using time-steppers: A reaction-diffusion example, Proc. Natl. Acad. Sci. USA, 97 (2000), pp. 9840-9843.
39. D. Xiu and G. Karniadakis, Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos, Comput. Methods Appl. Mech. Engrg., 191 (2002), pp. 4927-4948.
40. D. Xiu and C. E. Karniadakis, The Wiener - Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput., 24 (2002), pp. 619-644.
41. D. Xiu and G. Karniadakis, Modeling uncertainty in flow simulations via generalized polynomial chaos, J. Comput. Phys., 187 (2003), pp. 137-167.
42. D. Xiu and G. Karniadakis, A new stochastic approach to transient heat conduction modeling with uncertainty, Inter. J. Heat Mass Trans., 46 (2003), pp. 4681-4693.