Micro-macro Fokker--Planck--Kolmogorov models for a gas of rigid spheres

S. V. Bogomolov ; N. B. Esikova ; A. E. Kuvshinnikov

Matematičeskoe modelirovanie, Tome 28 (2016) no. 2, pp. 65-85.

Résumé

Macroscopic system of gas dynamic equations, differing from Navier–Stokes and quasi gas dynamic ones, is derived from a stochastic microscopic model of a hard sphere gas in a phase space. The model is diffusive in velocity space and valid for moderate Knudsen numbers. The main pecularity of our derivation is more accurate velocity averaging due to analitical solving stochastic differential equations with respect to Wiener mesure which describe our original meso model. It is shown at an example of a shock wave front structure that our approach leads to larger than Navier–Stokes front widening that corresponds to reality. The numerical solution is performed by a well suited to super computer applications special «discontinious» particle method.

Détail

Keywords: Boltzmann equation, Navier–Stokes equation; random processes, stochastic differential equations with respect to Poisson and Wiener measures, particle method.
Mots-clés : Kolmogorov–Fokker–Planck equation

Bibliographie

[1] O. M. Belotserkovskii, V. E. Ianitskii, “Statisticheskii metod chastits v iacheikakh dlia resheniia zadach dinamiki razrezhennogo gaza. I: Osnvy postroeniia metoda”, G. vychisl. matem. i matem. fiz., 15:5 (1975), 1195–1208 | MR

[2] O. M. Belotserkovskii, N. N. Fimin, V. M. Chechetkin, “Kogerentnye gidrodinamicheskie struktury i vikhrevaia dinamika”, Matem. modelirovanie, 27:8 (2015), 63–84

[3] S. A. Zabelok, R. R. Arslanbekov, V. I. Kolobov, “Adaptive kinetic-fluid solvers for heterogeneous computing architectures”, Journal of Computational Physics, 303 (2015), 455–469 | DOI | MR

[4] V. I. Bogachev, N. V. Krylov, M. Rekner, S. V. Shaposhnikov, Uravnenia Fokkera–Planka–Kolmogorova, IKI, 2013, 576 pp.

[5] Ya. B. Zeldovich, S. A. Molchanov, A. A. Ruzmaikin, D. D. Sokolov, “Peremezhaemost v sluchainoi srede”, UFN, 152:1 (1987), 3–32 | DOI | MR

[6] M. H. Gorji, P. Jenny, “Fokker–Planck-DSMC algorithm for simulations of rarefied gas flows”, Journal of Computational Physics, 287 (2015), 110–129 | DOI | MR

[7] J. Zhang, D. Zeng, J. Fan, “Analysis of transport properties determined by Langevin dynamics using Green–Kubo formulae”, Physica A: Statistical Mechanics and its Applications, 411 (2014), 104–112 | DOI | MR

[8] M. F. Ivanov, V. A. Galburt, “Stokhasticheskii podkhod k chislennomu resheniiu uravneniia Fokkera–Planka”, Matem. modelirovanie, 20:11 (2008), 3–27 | MR | Zbl

[9] V. K. Gupta, M. Torrilhon, “Comparison of relaxation phenomena in binary gas-mixtures of Maxwell molecules and hard spheres”, Computers Mathematics with Applications, 70:1 (2015), 73–88 | DOI | MR

[10] S. V. Bogomolov, “An approach to deriving stochastic gas dynamics models”, Doklady Mathematics, 78 (2008), 929–931 | DOI | MR | Zbl

[11] A. A. Arsen'yev, “On the approximation of the solution of the Boltzmann equation by solutions of the ito stochastic differential equations”, USSR Computational Mathematics and Mathematical Physics, 27:2 (1987), 51–59 | DOI | MR | Zbl

[12] S. V. Bogomolov, L. V. Dorodnitsyn, “Equations of stochastic quasi-gas dynamics: Viscous gas case”, Mathematical Models and Computer Simulations, 3:4 (2011), 457–467 | DOI | MR | Zbl

[13] S. V. Bogomolov, “Stochastic quasi gas dynamics equations as a base for particle methods”, Proc. V European Conference on Computational Fluid Dynamics ECCOMAS CFD 2010 (Lisbon, Portugal, 2010)

[14] S. Chandrasekhar, “Stochastic problems in physics and astronomy”, Rev. Modern. Phys., 15:1 (1943), 1–89 | DOI | MR | Zbl

[15] J. G. Kirkwood, “The statistical mechanical theory of transport processes”, J. Chem. Phys., 14:3 (1946), 180–201 | DOI

[16] C. Cercignani, Rarefied Gas Dynamics: From Basic Concepts to Actual Calculations, Cambridge University Press, 2000 | MR | Zbl

[17] A. V. Skorokhod, Stochastic Equations for Complex Systems, Kluwer Academic, Dordrecht, 1987 | MR | MR

[18] S. V. Bogomolov, I. G. Gudich, “Verification of a stochastic diffusion gas model”, Mathematical Models and Computer Simulations, 6:3 (2014), 305–316 | DOI | MR | MR

[19] B. N. Chetverushkin, “Resolution limits of continuous media mode and their mathematical formulations”, Mathematical Models and Computer Simulations, 5:3 (2013), 266–279 | DOI | MR

[20] T. G. Elizarova, Quasi-Gas Dynamic Equations, Springer, 2009, 283 pp. | MR | Zbl

[21] H. Brenner, “Bi-velocity hydrodynamics”, Physica A: Statistical Mechanics and its Applications, 388 (2009), 3391–3398 | DOI

[22] J. Mathiaud, L. Mieussens, A Fokker–Planck model of the Boltzmann equation with correct Prandtl number, 2015, arXiv: math.AP/1503.01246 | MR

[23] K. Morinishi, “A continuum/kinetic hybrid approach for multi-scale flow”, Proc. ECCOMAS CFD 2006 (Egmond aan Zee, Netherlands, 2006)

[24] A. K. Aringazin, M. I. Mazhintov, “Stochastic Models of Lagrangian Acceleration of Fluid Particle in Developed Turbulence”, Int J. Mod. Phys. B, 18 (2004), 3095–3168 | DOI | Zbl

[25] B. Oksendal, Stochastic Differental Equations, 6th edition, Springer, Berlin, 2000, 390 pp.

[26] S. K. Dadzie, J. M. Reese, A Fokker–Planck model of the Boltzmann equation with correct Prandtl number, 2011, arXiv: math-ph/1202.3169

[27] G. Röpke, Statistische Mechanik für das Nichtgleichgewicht, Deutscher Verlag der Wissenschaften, Berlin, 1987 | MR | MR

[28] S. V. Bogomolov, “An Entropy Consistent Particle Method for Navier–Stokes Equations”, Proc. IV European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2004 (Jyvaskyla, Finland, 2004)

[29] S. V. Bogomolov, K. V. Kuznetsov, “Metod chastits dlya sistemy uravnenii gazovoi dinamiki”, Matematicheskoe modelirovanie, 10:7 (1998), 93–100 | MR | Zbl

[30] H. Alsmeyer, “Density profiles in argon and nitrogen shock waves measured by the absorption of an electronic beam”, J. Fluid Mech., 74:3 (1976), 497–513 | DOI

[31] S. S. Stepanov, Stochastic world, Springer International Publishing, 2013 | MR | Zbl