Abstract
The article reports some results of test calculations of one model from the hierarchical set of models of gas dynamics obtained (a brief scheme of the deduction is presented) from a system of stochastic differential equations describing gas at moderate and small Knudsen numbers.
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References
A. V. Skorokhod, Stochastic Equations for Complicated Systems (Nauka, Moscow, 1983) [in Russian].
A. A. Arsen’ev, Lectures on Kinetic Euqations (Nauka, Moscow, 1992) [in Russian].
K. Cherchin’yani, Theory and Applications of the Boltzmann Equations (Mir, Moscow, 1978) [in Russian].
C. Cercignani, Rarefied Gas Dynamics (University Press, Cambridge, 2000).
B. N. Chetverushkin, The Kinetic Schemes and Quasi-gas-dynamic System of Euqations (MAKS Press, Moscow, 2004) [in Russian].
Yu. L. Klimontovich, “The Need and Possibility for a Unified Description of Kinetic and Hydrodynamic Processes,” Teor. i matem. fizika 92(2), 312 (1992).
G. Repke, Non-equilibrium Statistical Mechanics (Mir, Moscow, 1990) [in Russian].
K. A. Oelschlager, “Martingale Approach to the Law of Large Number for Weakly Interacting Stochastic Processes,” The Annals of Probability 12 (2), 458 (1984).
V. I. Kolobov, R. R. Arslanbekov, V. V. Aristov, et al., “Unified Solver for Rarefied and Continuum Flows with Adaptive Mesh and Algorithm Refinement,” Journal of Computational Physics 223, 589 (2007).
K. Morinishi, “A Continuum/Kinetic Hybrid Approach for Multi — Scale Flow Simulation,” European Conference on Computational Fluid Dynamics ECCOMAS CFD 2006 (TU Delft, the Netherlands).
P. Degond and M. Lemou, “Turbulence Models for Incompressible Fluids Derived from Kinetic Theory,” J. Math. Fluid Mech. 257 (2002).
T. G. Elizarova, Quasi-gas-dynamic Euqations and Method for Calculation of Viscous Flows (Nauchnyi Mir, Moscow, 2007) [in Russian].
A. Lukschin, H. Neunzert, and J. Struckmeier, “Coupling of Navier — Stokes and Boltzmann Regions,” (HER-MES Aerodynamics R/Q Program meeting, VKI, 1992).
S. V. Bogomolov, “Stochastic Model of Hydrodynamics,” Mat. Mod. 2(11), 85 (1990).
S. V. Bogomolov, “The Fokker-Planck Equation for Gas With the Knudsen Moderate Numbers,” Mat. Mod. 15(4), 16 (2003).
S.V. Bogomolov, “On Fokker-Planck Model for the Integral of the Boltzmann Collisions with the Moderate Knudsen Numbers,” Mat. Mod. 21(1), 111–117 (2009).
S. V. Bogomolov, “An Approach to Derivation of Stochastic Models of Gas Dynamics,” DAN 423(4), 458 (2008).
S. V. Bogomolov, “Quasi-dynamic Equations,” Mat. Mod. 21(12), 145 (2009).
Yu.V. Sheretov. Dinamika sploshnykh sred pri prostranstvenno-vremennom osrednenii. — (Izhevsk, Moscow) [in Russian].
I. A. Ivakhnenko, S.V. Polyakov, and B.N. Chetverushkin, “Quasi-hydro-dynamic Model and Small-Scale Turbulence,” Mat. Mod. 1(1), pp. 44–50 (2009).
H. Brenner, “Fluid Mechanics Revisited,” Physica A 370, pp. 190–224 (2006).
H. Brenner, “Bi-velocity hydrodynamics,” Physica A 388, pp. 3391–3398 (2009).
A. Bardow and H.C. Oettinger, “Consequences of the Brenner Modification to the Navier-Stokes Equations for Dynamic Light Scattering,” Physica A 373, pp. 88–96 (2007).
S. K. Dadzie, J. M. Reese, and C. R. McInnes, “A Continuum Model of Gas Flows with Localized Density Variations,” Physica A 387, pp. 6079–6094 (2008).
B. Oksendal’, Stochastic Differential Equations (Mir, Moscow, 2003) [in Russian].
G. Berd, Molecular Gas Dynamics (Mir, Moscow, 1981) [in Russian].
U. Ghia, K. N. Ghia, C.T. Shin, “High-Re Solutions for Incompressible Flow Using the Navier-Stokes Equations and a Multigrid Method,” J. Comput. Physics 48(3), pp. 387–411 (1982).
L. V. Dorodnitsyn and B. N. Chetverushkin, “An Implicit Scheme for Simulation of Subsonic Gas Flow,” Mat. Mod. 9(5). pp. 108–118 (1997).
A. A. Samarskii, Theory of Differential Schemes (Nauka, Moscow, 1983) [in Russian].
L. V. Dorodnitsyn, “Nonreflecting Boundary Conditions For Gas-dynamic Problems in the Nonlinear Statement,” Applied Mathematics and Informatics No.11 (MAKS Press, Moscow, 2002) [in Russian].
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Original Russian Text © S.V. Bogomolov, L.V. Dorodnitsyn, 2010, published in Matematicheskoe Modelirovanie, 2010, Vol. 22, No. 12, pp. 49–64.
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Bogomolov, S.V., Dorodnitsyn, L.V. Equations of stochastic quasi-gas dynamics: Viscous gas case. Math Models Comput Simul 3, 457–467 (2011). https://doi.org/10.1134/S207004821104003X
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DOI: https://doi.org/10.1134/S207004821104003X