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Hydrodynamic Regimes, Knudsen Layer, Numerical Schemes: Definition of Boundary Fluxes

Published online by Cambridge University Press:  03 June 2015

Christophe Besse ,
Saja Borghol ,
Thierry Goudon ,
Ingrid Lacroix-Violet  and
Jean-Paul Dudon
Christophe Besse*
Affiliation:
Project-Team SIMPAF, INRIA Lille Nord Europe Research Center, Park Plazza, 40 av. Halley, 59650 Villeneuve d’Ascq France Labo P. Painlevé UMR 8524 CNRS-Université des Sciences et Technologies Lille
Saja Borghol*
Affiliation:
Project-Team SIMPAF, INRIA Lille Nord Europe Research Center, Park Plazza, 40 av. Halley, 59650 Villeneuve d’Ascq France Labo P. Painlevé UMR 8524 CNRS-Université des Sciences et Technologies Lille
Thierry Goudon*
Affiliation:
Project-Team SIMPAF, INRIA Lille Nord Europe Research Center, Park Plazza, 40 av. Halley, 59650 Villeneuve d’Ascq France Labo P. Painlevé UMR 8524 CNRS-Université des Sciences et Technologies Lille
Ingrid Lacroix-Violet*
Affiliation:
Project-Team SIMPAF, INRIA Lille Nord Europe Research Center, Park Plazza, 40 av. Halley, 59650 Villeneuve d’Ascq France Labo P. Painlevé UMR 8524 CNRS-Université des Sciences et Technologies Lille
Jean-Paul Dudon*
Affiliation:
Thales Alenia Space, Cannes La Bocca
*
URL: http://math.univ-lille1.fr/∼goudon/ Email: christophe.besse@math.univ-lille1.fr
Email: saja.borghol@math.univ-lille1.fr
Corresponding author. Email: thierry.goudon@inria.fr
Email: ingrid.violet@math.univ-lille1.fr
Email: jean-paul.dudon@thalesaleniaspace.com
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Abstract

We propose a numerical solution to incorporate in the simulation of a system of conservation laws boundary conditions that come from a microscopic modeling in the small mean free path regime. The typical example we discuss is the derivation of the Euler system from the BGK equation. The boundary condition relies on the analysis of boundary layers formation that accounts from the fact that the incoming kinetic flux might be far from the thermodynamic equilibrium.

Keywords

35L6535Q3582C8082C4076M1265M08Hydrodynamic regimesKnudsen layerfinite volume schemeinitial-boundary value problems for conservation lawsEvaporation-condensation problem
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 3 , Issue 5 , October 2011 , pp. 519 - 561
Copyright
Copyright © Global-Science Press 2011

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References

[1] Aoki, K. and Masukawa, N., Gas flows caused by evaporation and condensation on two parallel condensed phases and the negative temperature gradient: numerical analysis by using a nonlinear kinetic equation, Phys. Fluids, 6(3) (1994), pp. 1379–1395.CrossRefGoogle Scholar
[2] Arnold, A. and Giering, U., An analysis of the Marshak conditions for matching Boltzmann and Euler equations, Math. Models Methods Appl. Sci., 7(4) (1997), pp. 557–577.Google Scholar
[3] Arthur, M. and Cercignani, C., Non-existence of a steady rarefied supersonic flow in a half-space, ZAMP., 31 (1980), pp. 635–645.Google Scholar
[4] Babovsky, H., Derivation of stochastic reflection laws from specular reflection, Trans. Theory Statist. Phys., 16(1) (1987), pp. 113–126.Google Scholar
[5] Bardos, C., Problémes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorémes d’approximation; application à l’équation de transport, Ann. Sci. école Norm. Sup., 3(4) (1970), pp. 185–233.CrossRefGoogle Scholar
[6] Bardos, C., Caflisch, R. and Nicolaenko, B., The Milne and Kramers problems for the Boltzmann equation of a hard sphere gas, Commun. Pure Appl. Math., 39 (1986), pp. 323–352.CrossRefGoogle Scholar
[7] Bardos, C., Golse, F. and Levermore, D., Fluid dynamic limits of kinetic equations I, formal derivations, J. Statist. Phys., 63(1-2) (1991), pp. 323–344.Google Scholar
[8] Bardos, C., Golse, F. and Levermore, D., Fluid dynamic limits of kinetic equations II, convergence proofs for the Boltzmann equation, Commun. Pure Appl. Math., 46(5) (1993), pp. 667–753.Google Scholar
[9] Bardos, C., Golse, F. and Sone, Y., Half-space problems for the Boltzmann equation: a survey, J. Stat. Phys., 124(2-4) (2006), pp. 275–300.Google Scholar
[10] Beals, R. and Protopopescu, V., Abstract time-dependent transport equations, J. Math. Anal. Appl., 121 (1987), pp. 370–405.CrossRefGoogle Scholar
[11] Bhatnagar, P. L., Gross, E. P. and Krook, M., 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.CrossRefGoogle Scholar
[12] Bourgat, J. F., Le Tallec, P., Perthame, B. and Qiu, Y., Coupling Boltzmann and Euler equations without overlapping, in Quarteroni, A., Periaux, J., Kutznetsov, Y. and Widlund, O. editors, Domain Decomposition Methods in Science and Engineering, The sixth international conference on domain decomposition, Como, Italy, June 1992, volume 157 of Contemp. Math., pp. 377–398. AMS, 1994.Google Scholar
[13] Cercignani, C., Half-space problems in the kinetic theory of gases, in Kräner, E. and Kirchgässner, K., editors, Trends to Applications of Pure Mathematics to Mechanics, volume 249 of Lecture Notes in Physics, pp. 35–51, Springer-Verlag, 1987.CrossRefGoogle Scholar
[14] Cercignani, C., The Boltzmann Equation and Its Applications, volume 67 of Applied Mathematical Sciences, Springer-Verlag, 1988.Google Scholar
[15] Cercignani, C., Scattering kernels for gas-surface interaction, in Proceedings of the Workshop on Hypersonic Flows for Reentry Problems, INRIA, Antibes, volume I, pp. 9–29, 1990.Google Scholar
[16] Cercignani, C., Lampis, M. and Lentati, A., A new scattering kernel in kinetic theory of gases, Trans. Theory Statist. Phys., 24(9) (1995), pp. 1319–1336.Google Scholar
[17] Cessenat, M., Théorémes de trace Lp pour des espaces de fonctions de la neutronique, C. R. Acad. Sci. Paris Sér. I Math., 299 (1984), pp. 831–834.Google Scholar
[18] Cessenat, M., Théorémes de trace pour des espaces de fonctions de la neutronique, C. R. Acad. Sci. Paris Sér. I Math., 300 (1985), pp. 89–92.Google Scholar
[19] Charles, F., Vauchelet, N., Besse, C., Goudon, T., Lacroix-Violet, I., Dudon, J.-P. and Navoret, L., Numerical approximation of Knudsen layesr for Euler-Poisson system, Tech. Rep., Cemracs 2010, 2011.Google Scholar
[20] Coron, F., Golse, F. and Sulem, C., A classification of well-posed kinetic layer problems, Commun. Pure Appl. Math., 41(4) (1988), pp. 409–435.CrossRefGoogle Scholar
[21] Darrozés, J.-S. and Guiraud, J.-P., Généralisation formelle du théoréme Hen présence de parois, C. R. Acad. Sci. Paris, 262 (1966), pp. 369–371.Google Scholar
[22] Dellacherie, S., Coupling of the Wang Chang-Uhlenbeck equations with the multispecies Euler system, J. Comput. Phys., 189 (2003), pp. 239–276.Google Scholar
[23] Dellacherie, S., Etude et discrétisation de modéles cinétiques et de modéles fluides à bas nombre de Mach, Habilitation à diriger des recherches, Univ. Paris 6, 2010.Google Scholar
[24] Diperna, R. J. and Lions, P.-L., On the Cauchy problem for Boltzmann equations: global existence and weak stability, Ann. Math., 130(2) (1989), pp. 321–366.Google Scholar
[25] Enquist, B. and Osher, S., Stable and entropy satisfying approximations for transonic flow calculations, Math. Comput., 31 (1981), pp. 45–75.Google Scholar
[26] Filbet, J. and Jin, S., A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources, J. Comput. Phys., 229(20) (2010), pp. 7625–7648.Google Scholar
[27] Friedrichs, K. O. and Lax, P. D., Systems of conservation equations with a convex extension, Pro. Nat. Acad. Sci. USA, 68 (1971), pp. 1686–1688.Google Scholar
[28] Godlewski, E. and Raviart, P.-A., Numerical approximation of hyperbolic systems of conservation laws, volume 118 of Applied Mathematical Sciences, Springer, New-York, 1996.Google Scholar
[29] Godounov, S. K., Lois de conservation et intégrales d’énergie des équations hyperboliques, in Carasso, C., Raviart, P.-A., and Serre, D., editors, Nonlinear hyperbolic problems (St. Etienne, 1986), volume 1270 of Lecture Notes in Math, pp. 135–149. Springer, Berlin, 1987.Google Scholar
[30] Golse, F., Applications of the Boltzmann equation within the context of upper atmosphere vehicle aerodynamics, Comput. Methods Appl. Mech. Eng., 75 (1989), pp. 299–316.Google Scholar
[31] Golse, F., Knudsen layers from a computational viewpoint, Trans. Theory Statist. Phys., 21(3) (1992), pp. 211–236.Google Scholar
[32] Golse, F., Boundary and interface layers for kinetic models, Tech. Rep., GdR SPARCH-CNRS, September 1997, Lecture Notes of the 4th Summer School of the GdR SPARCH, St Pierre d’Oléron.Google Scholar
[33] Golse, F. and Klar, A., A numerical method for computing asymptotic states and outgoing distributions for kinetic linear half-space problems, J. Statist. Phys., 80(5-6) (1995), pp. 1033–1061.Google Scholar
[34] Golse, F. and Poupaud, F., Stationary solutions of the linearized Boltzmann equation in a half-space, Math. Meth. Appl. Sci., 11 (1989), pp. 486–502.Google Scholar
[35] Golse, F. and Saint-Raymond, L., The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels, Invent. Math., 155(1) (2004), pp. 81–161.Google Scholar
[36] Goudon, T., Existence of solutions of transport equations with nonlinear boundary conditions, Euro. J. Mech. B Fluids, 16(4) (1997), pp. 557–574.Google Scholar
[37] Goudon, T., Sur Quelques Questions Relatives à la Théorie Cinétique des Gaz et à l’équation de Boltzmann, PhD thesis, Université Bordeaux 1, 1997.Google Scholar
[38] Greenberg, W. and Van der Mee, C. V. M., An abstract approach to evaporation models in rarefied gas dynamics, Z. Angew. Math. Phys., 35(2) (1984), pp. 156–165.Google Scholar
[39] Hamdache, K., Initial-boundary value problems for the Boltzmann equation: global existence of weak solutions, Arch. Rational Mech. Anal., 119(4) (1992), pp. 309–353.Google Scholar
[40] Jin, S., Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput., 21(2) (1999), pp. 441–454.Google Scholar
[41] Klar, A., Domain decomposition for kinetic problems with nonequilibrium states, Euro. J. Mech. B Fluids, 15(2) (1996), pp. 203–216.Google Scholar
[42] Kuscer, I., Phenomenological aspects of gas-surface interaction, in Cohen, E. and Fiszdon, W., editors, Fundamental Problems in Statistical Mechanics, volume IV, pp. 441–467, 1978.Google Scholar
[43] Leveque, R. J., Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002.Google Scholar
[44] Marshak, R. E., Note on the spherical harmonic method as applied to the Milne problem for a sphere, Phys. Rev., 71 (1947), pp. 443–446.Google Scholar
[45] Maxwell, J. C., On stresses in rarefied gases arising from inequalities of temperature, Phil. Trans. Royal Soc. London, 170 (1879), pp. 231–256.Google Scholar
[46] Mischler, S., On the trace problem for solutions of the Vlasov equation, Commun. Partial Differential Equations, 25(7-8) (2000), pp. 1415–1443.Google Scholar
[47] Mischler, S., Kinetic equations with Maxwell boundary condition, Tech. Rep., Univ. Paris Dauphine, 2010, hal-00346628.Google Scholar
[48] Perthame, B., Global existence to the BGK model of Boltzmann equation, J. Differential Equations, 82(1) (1989), pp. 191–205.Google Scholar
[49] Perthame, B., Second order Boltzmann schemes for compressible Euler equations in one and two space dimensions, SIAM J. Numer. Anal., 29(1) (1992), pp. 1–19.Google Scholar
[50] Perthame, B., Kinetic formulation of conservation laws, volume 21 of Oxford Lecture Series in Math and its Application, Oxford Univ. Press, 2002.Google Scholar
[51] Perthame, B. and Pulvirenti, M., Weighted L∞ bounds and uniqueness for the Boltzmann BGK model, Arch. Rational Mech. Anal., 125(3) (1993), pp. 289–295.Google Scholar
[52] Serre, D., Systems of Conservation Laws: Volume 1: Hyperbolicity, Entropies, Shock Waves. Cambridge University Press, Cambridge, 1999.Google Scholar
[53] Serre, D., Systems of Conservation Laws: Volume 2: Geometric Structures, Oscillation and Mixed Problems, Cambridge University Press, Cambridge, 2000.Google Scholar
[54] Sone, Y., Kinetic theory and fluid dynamics, Modeling and Simulation in science, Engineering and Technology, Birkhauser, 2002.Google Scholar
[55] Sone, Y., Ohwada, T. and Aoki, K., Evaporation and condensation on a plane condensed phase: numerical analysis of the linearized Boltzmann equation for hard-sphere molecules, Phys. Fluids, 1(8) (1989), pp. 1398–1405.Google Scholar
[56] Vasseur, A., Rigorous derivation of the kinetic/fluid coupling involving a kinetic layer on a toy problem, Archiv. Rat. Mech. Anal., 2010, to appear.Google Scholar
[57] Villani, C., Limites hydrodynamiques de l’équation de Boltzmann (d’aprés C. Bardos, F. Golse, C. D.Levermore, P.-L. Lions, N. Masmoudi, L. Saint-Raymond), Astérisque, (282): Exp. No.893, ix, pp. 365–405, 2002, Séminaire Bourbaki, Vol. 2000/2001.Google Scholar
[58] Wang, L., Liu, J. and Jin, S., Domain decomposition method for a two-scale hyperbolic system, Tech. Rep., UW-Madison, 2011.Google Scholar