Numerical properties of high order discrete velocity solutions to the BGK kinetic equation

Applied Numerical Mathematics

Volumn 61, Issue 4, 2011, Pages 410-427

  Alekseenko, A M ^a  

^a CALIFORNIA STATE UNIVERSITY   (United States)

Author keywords

Discontinuous Galerkin methods; Heat transfer; Kinetic equations; Normal shock wave; Transient gas flows

Indexed keywords

DIFFERENT ORDER; DISCONTINUOUS GALERKIN; DISCONTINUOUS GALERKIN METHODS; DISCRETIZATIONS; HIGH ORDER; HIGH-ORDER NUMERICAL METHOD; KINETIC EQUATIONS; MODEL KINETICS; NORMAL SHOCK WAVES; NUMERICAL PROPERTIES; NUMERICAL SOLUTION; ONE-DIMENSIONAL HEAT; RUNGE-KUTTA; SYNTHETIC MODELS; TEST ACCURACY;

CONVERGENCE OF NUMERICAL METHODS; FLOW OF GASES; HEAT TRANSFER; INTEGRAL EQUATIONS; KINETIC ENERGY; KINETIC THEORY; KINETICS; RUNGE KUTTA METHODS; SHOCK WAVES; WAVE EQUATIONS; WAVES;

GALERKIN METHODS;

EID: 78651342454     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apnum.2010.11.005     Document Type: Article

References (40)

1. A. Alexeenko, C. Galitzine, and A.M. Alekseenko High-order discontinuous Galerkin method for Boltzmann model equations in: 40th AIAA Thermophysics Conference, Seattle, WA http://www.aiaa.org/agenda.cfm?lumeetingid= 1820P&dateget=25-Jun-08#session9731 2008
2. P. Andries, K. Aoki, and B. Perthame A consistent BGK-type model for gas mixtures http://hal.inria.fr/inria-00072389/en/
3. P. Andries, J.-F. Bourgat, P. le Tallec, and B. Perthame Numerical comparison between the Boltzmann and ES-BGK models for rarefied gases Comput. Methods Appl. Mech. Engrg. 191 31 2002 3369 3390
4. P. Andries, P. le Tallec, J.-P. Perlat, and B. Perthame The Gaussian-BGK model of Boltzmann equation with small Prandtl number Eur. J. Mech. B Fluids 19 6 2000 813 830
5. P.L. Bhatnagar, 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 3 1954 511 525
6. G. Bird Molecular Gas Dynamics and the Direct Simulation of Gas Flow Oxford Engineering Science Series 1994 Clarendon
7. J.A. Carrillo, I.M. Gamba, A. Majorana, and C.-W. Shu A WENO-solver for the transients of Boltzmann-Poisson system for semiconductor devices: performance and comparisons with Monte Carlo methods J. Comput. Phys. 184 2 2003 498 525
8. C. Cercignani The Boltzmann Equation and its Applications Applied Mathematical Sciences vol. 67 1988 Springer-Verlag New York
9. S. Chen, W. E, Y. Liu, and C.-W. Shu A discontinuous Galerkin implementation of a domain decomposition method for kinetic-hydrodynamic coupling multiscale problems in gas dynamics and device simulations J. Comput. Phys. 225 2 2007 1314 1330 http://www.sciencedirect.com/science/article/B6WHY- 4N0HJKH-3/2/1a286ba61fbc1e882e65ec7a5400dc22
10. Y. Cheng, I.M. Gamba, A. Majorana, and C.-W. Shu A discontinuous Galerkin solver for Boltzmann-Poisson systems in nano devices Comput. Methods Appl. Mech. Engrg. 198 37-40 2009 3130 3150 http://www.sciencedirect.com/science/ article/B6V29-4WFG5PG-4/2/3b6e2295a1d5ff9c8d8f7881db6221b6
11. B. Cockburn Discontinuous Galerkin methods for convection-dominated problems T. Barth, H. Deconink, High-order Methods for Computational Physics Lect. Notes Comput. Sci. Eng. vol. 9 1999 Springer-Verlag Berlin 69 224
12. B. Cockburn, and C.-W. Shu Runge-Kutta discontinuous Galerkin methods for convection-dominated problems J. Sci. Comput. 16 3 2001 173 261 (Pubitemid 33818237)
13. L.V. Fausett Applied Numerical Analysis Using MATLAB 1999 Prentice Hall
14. Y. Cheng, I.M. Gamba, J. Proft, Positivity-preserving discontinuous Galerkin schemes for linear Vlasov-Boltzmann transport equations, Mathematics of Computation (2010), in press.
15. M.K. Gobbert, and T.S. Cale A Galerkin method for the simulation of the transient 2-D/2-D and 3-D/3-D linear Boltzmann equation J. Sci. Comput. 30 2 2007 237 273
16. J.M. Haile Molecular Dynamics Simulation: Elementary Methods 1997 Wiley
17. L.H. Holway New statistical models for kinetic theory: methods of construction Phys. Fluids 9 9 1966 1658 1673
18. B. Huang, and D.P. Giddens The discrete ordinate method for the linearized boundary value problems in kinetic theory of gases C. Brundin, Rarefied Gas Dynamics vol. 1 1967 Academic Press New York 481 504
19. M.S. Ivanov, and S.F. Gimelshein Computational hypersonic rarefied flows Annual Rev. Fluid Mech. 30 1 1998 469 505 http://arjournals.annualreviews.org/ doi/abs/10.1146/annurev.fluid.30.1.469
20. E. Josyula, R. Arslanbekov, V. Kolobov, S. Gimelshein, Testing of the unified flow solver (UFS) for nozzle and plume flows, AIAA paper 2007-209.
21. C. Kim, K. Xu, L. Martinelli, and A. Jameson Analysis and implementation of the gas-kinetic BGK scheme for computational gas dynamics Internat. J. Numer. Methods Fluids 25 1 1997 21 49
22. S. Li, and W.K. Liu Meshfree and particle methods and their applications Appl. Mech. Rev. 55 2002 1 34
23. H. Liu, and K. Xu A Runge-Kutta discontinuous Galerkin method for viscous flow equations J. Comput. Phys. 224 2 2007 1223 1242
24. M.N. Macrossan ν-DSMC: A fast simulation method for rarefied flow J. Comput. Phys. 173 2 2001 600 619 http://www.sciencedirect.com/science/article/ B6WHY-45BC24W-1V/2/272ec05258e1fedde19f6ce80a4b3224
25. L. Mieussens Discrete-velocity models and numerical schemes for the Boltzmann-BGK equation in plane and axisymmetric geometries J. Comput. Phys. 162 2000 429 466
26. L. Mieussens, and H. Struchtrup Numerical comparison of Bhatnagar-Gross-Krook models with proper Prandtl number Phys. Fluids 16 8 2004 2797 2813
27. E.S. Oran, C.K. Oh, and B.Z. Cybyk Direct simulation Monte Carlo: Recent advances and applications Annual Rev. Fluid Mech. 30 1 1998 403 441 http://arjournals.annualreviews.org/doi/abs/10.1146/annurev.fluid.30.1.403
28. B. Perthame Global existence to the BGK model of Boltzmann equation J. Differential Equations 82 1 1989 191 205
29. B. Perthame Mathematical tools for kinetic equations Bull. Amer. Math. Soc. (NS) 41 2 2004 205 244 (electronic)
30. A. Ralston, and P. Rabinovich A First Course in Numerical Analysis 1978 McGraw-Hill New York
31. J.M. Reese, M.A. Gallis, and D.A. Lockerby New directions in fluid dynamics: non-equilibrium aerodynamic and microsystem flows R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 361 1813 2003 2967 2988 (mathematics, physics and engineering)
32. G. Richter On the convergence of the discrete ordinate method in kinetic theory SIAM J. Appl. Math. 25 2 1973 149 157
33. D.H. Rothman, and S. Zaleski Lattice-Gas Cellular Automata: Simple Models of Complex Hydrodynamics 1997 Cambridge University Press Cambridge, UK
34. S. Roy, R. Raju, H.F. Chuang, B. Cruden, and M. Meyyappan Modeling gas flow through microchannels and nanopores J. Appl. Phys. 93 8 2003 4870 4879
35. L. Saint-Raymond From the BGK model to the Navier-Stokes equations Ann. Sci. École Norm. Sup. 36 2 2003 271 317 http://www.sciencedirect.com/ science/article/B6VKH-48HS9DK-5/2/4b7102c9ed9f501112dc9b08b7c9ae3d
36. N. Selden, C. Ngalande, N. Gimelshein, S. Gimelshein, and A. Ketsdever Origins of radiometric forces on a circular vane with a temperature gradient J. Fluid Mech. 634 2009 419 431 10.1017/S0022112009007976
37. C.-W. Shu, and S. Osher Efficient implementation of essentially non-oscillatory shock-capturing schemes J. Comput. Phys. 77 1988 439 471
38. C.-W. Shu, and S. Osher Efficient implementation of essentially non-oscillatory shock-capturing schemes, II J. Comput. Phys. 83 1989 32 78
39. Y. Sone Molecular Gas Dynamics, Modeling and Simulation in Science, Engineering and Technology 2007 Birkhäuser Boston Inc. Boston, MA (theory, techniques, and applications)
40. D.C. Wadsworth, N.E. Gimelshein, S.F. Gimelshein, I.J. Wysong, Assessment of translational anisotropy in rarefied flows using kinetic approaches, in: American Institute of Physics Conference Series, vol. 1084, 2008, pp. 206-211.