A linearization of Mieussens’s discrete velocity model for kinetic equations. (English) Zbl 1160.82341

Summary: A linearization is developed for Mieussens’s discrete velocity model [see, e.g., L. Mieussens, J. Comput. Phys. 162, No. 2, 429–466 (2000; Zbl 0984.76070)] for kinetic equations. The basic idea is to use a linearized expression of the reference distribution function in the kinetic equation, instead of its exact expression, in the numerical scheme. This modified scheme is applied to various kinetic models, which include the BGK model, the ES-BGK model, the BGK model with velocity-dependent collision frequency, and the recently proposed ES-BGK model with velocity-dependent collision frequency. One-dimensional stationary shock waves and stationary planar Couette flow, which are two benchmark problems for rarefied gas flows, are chosen as test examples. Molecules are modeled as Maxwell molecules and hard sphere molecules. It is found that results from the modified scheme are very similar to results from the original Mieussens’s numerical scheme for various kinetic equations in almost all tests we did, while, depending on the test case, 20–40 percent of computational time can be saved. The application of the method is not affected by the Knudsen number and molecular models, but is restricted to lower Mach numbers for the BGK (or the ES-BGK) model with velocity-dependent collision frequency.


82C40 Kinetic theory of gases in time-dependent statistical mechanics
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics


Zbl 0984.76070
Full Text: DOI


[1] Struchtrup, H., Macroscopic transport equations for rarefied gas flows—approximation methods in kinetic theory, Interaction of mechanics and mathematics series, (2005), Springer Heidelberg · Zbl 1119.76002
[2] Kogan, M.N., Rarefied gas dynamics, (1969), Plenum Press
[3] Cercignani, C., Rarefied gas dynamics: from basic concepts to actual calculations, (2000), Cambridge University Press · Zbl 0961.76002
[4] Garzo, V.; Santos, A., Kinetic theory of gases in shear flows, (2003), Kluwer Academic Publishers Dordrecht · Zbl 1140.82030
[5] Y. Zheng, Analysis of kinetic models and macroscopic continuum equations for rarefied gas dynamics, Ph.D. thesis, Dept. Mech. Eng., Univ. of Victoria, Canada, 2004
[6] Mieussens, L., Discrete-velocity models and numerical schemes for the Boltzmann-BGK equation in plane and axisymmetric geometries, J. comput. phys., 162, 429-466, (2000) · Zbl 0984.76070
[7] Mieussens, L., Discrete velocity model and implicit scheme for the BGK equation of rarefied gas dynamics, Math. models methods appl. sci., 10, 1121-1149, (2000) · Zbl 1174.82329
[8] Mieussens, L., Convergence of a discrete-velocity model for the Boltzmann-BGK equation, Comput. math. appl., 41, 83-96, (2001) · Zbl 0980.82027
[9] Mieussens, L.; Struchtrup, H., Numerical comparison of BGK-models with proper Prandtl number, Phys. fluids, 16, 2797-2813, (2004) · Zbl 1186.76372
[10] Zheng, Y.; Struchtrup, H., Ellipsoidal statistical BGK model with velocity-dependent collision frequency, Phys. fluids, 17, 127103, (2005), (1-17) · Zbl 1188.76184
[11] Bhatnagar, P.L.; Gross, E.P.; Krook, M., A model for collision processes in gases. I: small amplitude processes in charged and neutral one-component systems, Phys. rev., 94, 511-525, (1954) · Zbl 0055.23609
[12] Lowell, J.; Holway, H., New statistical models for kinetic theory: methods of construction, Phys. fluids, 9, 1658-1673, (1966)
[13] Andries, P.; Perthame, B., The ES-BGK model equation with correct Prandtl number, (), 30-36
[14] Zheng, Y.; Struchtrup, H., Burnett equations for the ellipsoidal statistical BGK model, Continuum mech. thermodyn., 16, 97-108, (2004) · Zbl 1067.76079
[15] Struchtrup, H., The BGK-model with velocity-dependent collision frequency, Continuum mech. thermodyn., 9, 23-31, (1997) · Zbl 0891.76079
[16] Bouchut, F.; Perthame, B., A BGK model for small Prandtl number in the navier – stokes approximation, J. stat. phys., 71, 191-207, (1993) · Zbl 0943.76500
[17] Barichello, L.B.; Bartz, A.C.R.; Camargo, M.; Siewert, C.E., The temperature-jump problem for a variable collision frequency model, Phys. fluids, 14, 382-391, (2002) · Zbl 1184.76046
[18] Cercignani, C., The method of elementary solutions for kinetic models with velocity-dependent collision frequency, Ann. phys., 40, 469-481, (1966)
[19] Cercignani, C., Knudsen layers: theory and experiment, (), 187-195
[20] Yee, H.C., Construction of explicit and implicit symmetric TVD schemes and their applications, J. comput. phys., 68, 151-179, (1987) · Zbl 0621.76026
[21] Press, W.H.; Flannery, B.P.; Teukolsky, S.A.; Vetterling, W.T., Numerical recipes, (1986), Cambridge University Press · Zbl 0587.65005
[22] Dennis, J.E.; Schnabel, R.B., Numerical methods for unconstrained optimization and nonlinear equations, (1983), Prentice-Hall · Zbl 0579.65058
[23] L. Mieussens (Toulouse, France), private communication, 2003
[24] Pham-Van-Diep, G.C.; Erwin, D.A.; Muntz, E.P., Testing continuum descriptions of low-Mach-number shock structures, J. fluid mech., 232, 403-413, (1991) · Zbl 0729.76589
[25] A. Schuetze, Direct simulation by Monte Carlo modeling Couette flow using dsmc1as.f: A user’s manual, Technical Report, Dept. Mech. Eng., Univ. of Victoria, Canada, 2003
[26] Bird, R.B.; Stewart, W.E.; Lightfoot, E.N., Transport phenomena, (2002), John Wiley & Sons
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