Resolution limits of continuous media mode and their mathematical formulations

B. N. Chetverushkin, “High-Performance Computing: Fundamental Problems in Industrial Applications, in Parallel, Distributed and Grid Computing for Engineering, Eds. by B.H. Topping and P. Ivanyi (Saxe-Coburg Publications, 2009), pp. 96–108.

B. N. Chetverushkin, “Applied Mathematics and Problems of Using High Performance Computing Systems,” Tr. Moskovskogo Fiz. Tekh. Inst., 3(4), 96 (2011), pp. 96–108.

O. A. Ladyzhenskaya, Mathematical Theory of Dynamics of Viscous Incompressible Fluid (Nauka, Moscow, 1970) [in Russian].

A. I. Koshelev and S. I. Chelnak, Solution Regularity of Some Boundary Value Problems for Quasilinear Elliptic and Parabolic Systems (Izd. St. Petersburg Univ., St. Petersburg, 2000) [in Russian].

G. M. Kobel’kov, “Existence of “Global” Solution for Ocean Dynamics Equations,” Dokl. Akad. Nauk 407(1), 457 (2006).

L. G. Loitsianskii, Fluid and Gas Mechanics (Nauka, Moscow, 1987) [in Russian].

A. A. Vlasov, Statistical Distribution Functions (Nauka, Moscow, 1966) [in Russian].

V. E. Golant, A. P. Zhilinskii, and I. E. Sakharov, Principles of Plasma Physics (Atomizdat, Moscow 1977).

J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids (Wiley, New York, 1954).

S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases (Cambridge University Press, New York, 1952).

R. L. Liboff, Introduction to the Theory of Kinetic Equations (Wiley, New York, 1969).

V. V. Vedenyapin, Boltzmann and Vlasov Kinetic Equations (Fizmatlit, Moscow, 2001) [in Russian].

M. I. Volchinskaya, A. N. Pavlov, and B. N. Chetverushkin, “A Scheme for Integrating Equations of Gas Dynamics,” Preprint No. 113, (Keldysh Inst. of Applied Mathematics, Russian Academy of Sciences, Moscow, 1983).

B. N. Chetverushkin, Kinetic Shemes and Quasi-Gas-Dynamic System of Equations (CIMNE, Barcelona, 2008).

T. G. Yelizarova and B. N. Chetverushkin, “Use of Kinetic Models for Calculating Gas Dynamic Flows,” in Mathematical Modeling. Processes in Nonlinear Media (Nauka, Moscow, 1986) [in Russian].

T. G. Yelizarova, Quasi-Gas-Dynamic Equations and Methods for Computing Viscous Flows (Nauchnyi mir, Moscow, 2007) [in Russian].

A. A. Zlotnik and B. N. Chetverushkin, “Parabolicity of the Quasi-Gasdynamic System of Equations, its Hyperbolic Second-Order Modification, and the Stability of Small Perturbations for Them,” Comput. Math. Math. Phys. 48(3), 420 (2008).

A. A. Samarskii, Theory of Difference Schemes (Nauka, Moscow, 1989) [in Russian].

A. A. Davydov, B. N. Chetverushkin, and E. V. Shil’nikov, “Simulating Flows of Incompressible and Weakly Compressible Fluids on Multicore Hybrid Computer Systems,” Comput. Math. Math. Phys. 50(12), 157 (2010).

G. R. McNamara and G. Zanetti, “Use of the Boltzmann Equation for Simulate Lattice Gas Automata,” II Phys. Rev. Lett. 56(14), 1505 (1986).

M. Tsutahara, N. Takada, and N. Kataoka, Lattice Gas and Lattice Boltzmann Methods? New Method in Computational Fluid Dynamics (Corona Publishing, Tokyo, 1999).

S. Succi, The Lattice Boltzmann Equation in Fluid Dynamics and Beyond (Claredon Press, Oxford, 2001).

E. Oñate, “Derivation of Stabilized Equations for Numerical Solution of Advective Diffusive Transport and Fluid Flow Problems,” Comput. Math. Appl. Mech. Eng. 151, 233 (1998).

E. Oñate and M. Manzan, “Stabilization techniques for finite element analysis and convection diffusion problems,” CIMNE, No. 183 (2000).

L. S. Basniev, I. N. Kochina, and V. M. Maksimov, Subsurface Hydromechanics (Nauka, Moscow, 1993) [in Russian].

N. S. Bakhvalov and G. P. Panasenko, Averaging of Processes in Periodic Media (Nauka, Moscow, 1984) [in Russian].

M. A. Trapeznikova, M. S. Belotserkovskaya, and B. N. Chetverushkin, “An Analog of Kinetically Consistent Schemes for the Simulation of the Filtration Problem,” Mat. Model. 14(10), 69 (2002).

D. N. Morozov, M. A. Trapeznikova, B. N. Chetverushkin, and N. G. Churbanova, “Application of Explicit Schemes for the Simulation of the Two-Phase Filtration Process,” Math. Models Comput. Simul. 1, 62 (2012).

W. Wieser, L’équation de Boltzmann avec Terme de Viscosité, Solution Globales sous des Hypotheses Faibles sur la Condition Initial. C.R. Acad. Sc., 298, ser. I.N.S. (1989).

Yu. V. Sheretov, Mathematical Modeling of Flows on the Basis of Quasi-Hydrodynamic and Quasi-Gasdynamic Equations (Tver Univ., Tver, 2000) [in Russian].