Macroscopic fluid models with localized kinetic upscaling effects

Multiscale Modeling and Simulation

Volumn 5, Issue 3, 2006, Pages 940-979

  Degond, Pierre ^a   Liu, Jian Guo ^b   Mieussens, Luc ^a  

^a INSTITUT DE MATHÉMATIQUES DE TOULOUSE   (France)

^b University of Maryland   (United States)

Author keywords

Diffusion approximation; Hydrodynamic approximation; Kinetic equation; Kinetic fluid coupling

Indexed keywords

APPROXIMATION THEORY; DIFFUSION; HYDRODYNAMICS; MATHEMATICAL MODELS; NUMERICAL METHODS;

DIFFUSION APPROXIMATION; HYDRODYNAMIC APPROXIMATION; KINETIC EQUATION; KINETIC-FLUID COUPLING;

COMPUTATIONAL FLUID DYNAMICS;

EID: 34548537653     PISSN: 15403459     EISSN: 15403467     Source Type: Journal    
DOI: 10.1137/060651574     Document Type: Article

References (40)

1. F. F. ABRAHAM, J. Q. BROUGHTON, N. BERNSTEIN, AND E. KAXIRAS, Spanning the continuum to quantum length scales in a dynamic simulation of brittle fracture, Europhys. Lett., 44 (1998), pp. 783-787.
2. G. BAL AND Y. MADAY, Coupling of transport and diffusion models in linear transport theory, M2AN Math. Model. Numer. Anal., 36 (2002), pp. 69-86.
3. P. L. BATHNAGAR, E. P. GROSS, AND M. KROOK, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), pp. 511-525.
4. G. A. BIRD, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford Engrg. Sci. Ser. 42, Clarendon Press, Oxford University Press, New York, 1995.
5. J.-F. BOURGAT, P. LE TALLEC, AND M. D. TIDRIRI, Coupling Boltzmann and Navier-Stokes equations by friction, J. Comput. Phys., 127 (1996), pp. 227-245.
6. W. CAI, M. DE KONING, V. V. BULATOV, AND S. YIP, Minimizing boundary reflections in coupled-domain simulations, Phys. Rev. Lett., 85 (2000), pp. 3213-3216.
7. C. CERCIGNANI, The Boltzmann Equation and Its Applications, Appl. Math. Sci. 67, Springer-Verlag, New York, 1988.
8. S. CHANDRASEKHAR, Radiative Transfer, Dover, New York, 1960.
9. S. CHEN, W. E, Y. LIU, AND C.-W. SHU, A Domain Decomposition Method for Kinetic-Hydrodynamic Multiscale Problems in Gas Dynamics and Device Simulations Using the Discontinuous Galerkin Method, preprint.
10. H. CHOI AND J.-G. LIU, The reconstruction of upwind fluxes for conservation laws: Its behavior in dynamic and steady state calculations, J. Comput. Phys., 144 (1998), pp. 237-256.
11. N. CROUSEILLES, P. DEGOND, AND M. LEMOU, A hybrid kinetic /fluid model for solving the gas dynamics Boltzmann-BGK equation, J. Comput. Phys., 199 (2004), pp. 776-808.
12. N. CROUSEILLES, P. DEGOND, AND M. LEMOU, A hybrid kinetic-fluid model for solving the Vlasov-BGK equation, J. Comput. Phys., 203 (2005), pp. 572-601.
13. P. DEGOND, G. DIMARCO, S. JIN, AND L. MIEUSSENS, A Dynamic Transition between Kinetic and Hydrodynamic Equations, manuscript.
14. P. DEGOND AND S. JIN, A smooth transition model between kinetic and diffusion equations, SIAM J. Numer. Anal., 42 (2005), pp. 2671-2687.
15. P. DEGOND, S. JIN, AND L. MIEUSSENS, A smooth transition model between kinetic and hydrodynamic equations, J. Comput. Phys., 209 (2005), pp. 665-694.
16. P. DEGOND AND C. SCHMEISER, Kinetic boundary layers and fluid-kinetic coupling in semiconductors, Transport Theory Statist. Phys., 28 (1999), pp. 31-55.
17. W. E AND B. ENGQUIST, The heterogeneous multiscale method, Commun. Math. Sci., 1 (2003), pp. 87-132.
18. W. E AND Z. HUANG, A dynamic atomistic-continuum method for the simulation of crystalline materials, J. Comput. Phys., 182 (2002), pp. 234-261.
19. F. GOLSE, S. JIN, AND C. D. LEVERMORE, A domain decomposition analysis for a two-scale linear transport problem, M2AN Math. Model. Numer. Anal., 37 (2003), pp. 869-892.
20. A. B. HUANG AND P. F. HWANG, Test of statistical models for gases with and without internal energy states, Phys. Fluids, 16 (1973), pp. 466-475.
21. G.-S. JIANG AND C.-W. SHU, Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126 (1996), pp. 202-228.
22. S. JIN AND Z. P. XIN, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math., 48 (1995), pp. 235-276.
23. A. KLAR, Asymptotic-induced domain decomposition methods for kinetic and drift diffusion semiconductor equations, SIAM J. Sci. Comput., 19 (1998), pp. 2032-2050.
24. A. KLAR, H. NEUNZERT, AND J. STRUCKMEIER, Transition from kinetic theory to macroscopic fluid equations: A problem for domain decomposition and a source for new algorithms, Transport Theory Statist. Phys., 29 (2000), pp. 93-106.
25. A. KLAR AND C. SCHMEISER, Numerical passage from radiative heat transfer to nonlinear diffusion models, Math. Models Methods Appl. Sci., 11 (2001), pp. 749-767.
26. A. KLAR AND N. SIEDOW, Boundary layers and domain decomposition for radiative heat transfer and diffusion equations: Applications to glass manufacturing process, European J. Appl. Math., 9 (1998), pp. 351-372.
27. C. D. LEVERMORE, W. J. MOROKOFF, AND B. T. NADIGA, Moment realizability and the validity of the Navier-Stokes equations for rarefied gas dynamics, Phys. Fluids, 10 (1998), pp. 3214-3226.
28. X. LI AND W. E, Multiscale modeling of the dynamics of solids at finite temperature, J. Mech. Phys. Solids, 53 (2005), pp. 1650-1685.
29. T.-P. LIU AND S.-H. YU, Boltzmann equation: Micro-macro decompositions and positivity of shock profiles, Comm. Math. Phys., 246 (2004), pp. 133-179.
30. X.-D. LIU, S. OSHER, AND T. CHAN, Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115 (1994), pp. 200-212.
31. J. C. MANDAL AND S. M. DESHPANDE, Kinetic flux vector splitting for Euler equations, Comput. & Fluids, 23 (1994), pp. 447-478.
32. T. OHSAWA AND T. OHWADA, Deterministic hybrid computation of rarefied gas flows, in Proceedings of the 23rd International Symposium on Rarefied Gas Dynamics, Vol. 663, American Institute of Physics, Melville, NY, 2003, pp. 931-938.
33. T. OHWADA, Structure of normal shock waves: Direct numerical analysis of the Boltzmann equation for hard-sphere molecules, Phys. Fluids A, 5 (1993), pp. 217-234.
34. B. PERTHAME, Second-order Boltzmann schemes for compressible Euler equations in one and two space dimensions, SIAM J. Numer. Anal., 29 (1992), pp. 1-19.
35. Y. QIU, Étude des équations d'Euler et de Boltzmann et de leur couplage. Application â la simulation numérique d'écoulements hypersoniques de gaz raréfiés, Institut National de Recherche en Informatique et en Automatique (INRIA), Rocquencourt, France, 1993, Thèse, Université Paris VI, Paris, 1993.
36. J. SCHNEIDER, Direct coupling of fluid and kinetic equations, Transport Theory Statist. Phys., 25 (1996), pp. 681-698.
37. P. LE TALLEC AND F. MALLINGER, Coupling Boltzmann and Navier-Stokes equations by half fluxes, J. Comput. Phys., 136 (1997), pp. 51-67.
38. S. TIWARI, Coupling of the Boltzmann and Euler equations with automatic domain decomposition, J. Comput. Phys., 144 (1998), pp. 710-726.
39. G. J. WAGNER, E. G. KARPOV, AND W. K. LIU, Molecular dynamics boundary conditions for regular crystal lattices, Comput. Methods Appl. Mech. Engrg., 193 (2004), pp. 1579-1601.
40. H. C. YEE, A Class of High-Resolution Explicit and Implicit Shock-Capturing Methods, Technical Report Lecture Series 1989-04, von Karman Institute for Fluid Dynamics, Rhode-Saint-Genèse, Belgium, 1989.