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September  2013, 6(3): 533-543. doi: 10.3934/krm.2013.6.533

On a regularization of the magnetic gas dynamics system of equations

1. 

CEA, DAM, DIF, F-91297 Arpajon

2. 

Department of Mathematics at Faculty of Economics Sciences, National Research University Higher School of Economics, Myasnitskaya 20, 101000 Moscow

Received  August 2012 Revised  January 2013 Published  May 2013

A brief derivation of a specific regularization for the magnetic gas dynamic system of equations is given in the case of general equations of gas state (in presence of a body force and a heat source). The entropy balance equation in two forms is also derived for the system. For a constant magnetic regularization parameter and under a standard condition on the heat source, we show that the entropy production rate is nonnegative.
Citation: Bernard Ducomet, Alexander Zlotnik. On a regularization of the magnetic gas dynamics system of equations. Kinetic and Related Models, 2013, 6 (3) : 533-543. doi: 10.3934/krm.2013.6.533
References:
[1]

B. N. Chetverushkin, "Kinetic Schemes and Quasi-Gas Dynamic System of Equations," CIMNE, Barcelona, 2008.

[2]

T. G. Elizarova., "Quasi-Gas Dynamic Equations," Computational Fluid and Solid Mechanics, Springer, Dordrecht, 2009. doi: 10.1007/978-3-642-00292-2.

[3]

T. G. Elizarova, Time averaging as an approximate technique for constructing quasi-gas-dynamic and quasi-hydrodynamic equations, Comp. Math. Math. Phys., 51 (2011), 1973-1982. doi: 10.1134/S0965542511110078.

[4]

T. G. Elizarova and S. D. Ustjugov, Quasi-gas dynamics algorithm to solve equations of magnetohydrodynamics. One-dimensional case, (in Russian) Keldysh Inst. Appl. Math. Moscow, preprint No. 1, (2011). Available from: http://www.keldysh.ru/papers/2011/source/prep2011_01.pdf.

[5]

T. G. Elizarova and S. D. Ustjugov, Quasi-gas dynamics algorithm to solve equations of magnetohydrodynamics. Multidimensional case, (in Russian) Keldysh Inst. Appl. Math. Moscow, preprint No. 30, (2011). Available from: http://www.keldysh.ru/papers/2011/source/prep2011_30.pdf.

[6]

A. C. Eringen and G. A. Maugin, "Electrodynamics of Continua II: Fluids and Complex Media," Springer-Verlag, New York, 1990.

[7]

E. Feireisl and A. Novotný, "Singular Limits in Thermodynamics of Viscous Fluids," Advances in Mathematical Fluid Mechanics, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8843-0.

[8]

I. A. Kvasnikov, "Thermodynamics and Statistical Physics, Vol. 1: Theory of Equilibrium Systems and Thermodynamics," (in Russian) Editorial URSS, Moscow, 2002.

[9]

L. D. Landau and E. M. Lifshitz, "Electrodynamics of Continuous Media," Pergamon Press, Oxford, 1960.

[10]

Yu. V. Sheretov, "Continuum Dynamics Under Spatiotemporal Averaging," (in Russian) RKhD, Moscow-Izhevsk, 2009.

[11]

A. A. Zlotnik, Classification of some modifications of the Euler system of equations, Doklady Math., 73 (2006), 302-306. doi: 10.1134/S1064562406020396.

[12]

A. A. Zlotnik, Energy equalities and estimates for barotropic quasi-gas-dynamic and quasi-hydrodynamic systems of equations, Comp. Maths. Math. Phys., 50 (2010), 310-321. doi: 10.1134/S0965542510020120.

[13]

A. A. Zlotnik, A quasi-gas-dynamic system of equations with general state equations, Doklady Math., 81 (2010), 312-316. doi: 10.1134/S1064562410020419.

[14]

A. A. Zlotnik, Linearized stability of equilibrium solutions to the quasi-gas-dynamic system of equations, Doklady Math., 82 (2010), 811-815. doi: 10.1134/S1064562410050352.

[15]

A. A. Zlotnik, On the quasi-gas-dynamic system of equations with general equations of state and a heat source, (in Russian) Math. Modelling, 22 (2010), 53-64.

[16]

A. A. Zlotnik, On construction of quasi-gas-dynamic systems of equations and the barotropic system with the potential body force, (in Russian) Math. Modelling, 24 (2012), 65-79.

[17]

A. A. Zlotnik and B. N. Chetverushkin, Parabolicity of the quasi-gas-dynamic system of equations, its hyperbolic second-order modification, and the stability of small perturbations for them, Comp. Maths. Math. Phys., 48 (2008), 420-446. doi: 10.1134/S0965542508030081.

[18]

A. Zlotnik and V. Gavrilin, On quasi-gas-dynamic system of equations with general equations of state and its application, Math. Modelling Anal., 16 (2011), 509-526. doi: 10.3846/13926292.2011.627382.

show all references

References:
[1]

B. N. Chetverushkin, "Kinetic Schemes and Quasi-Gas Dynamic System of Equations," CIMNE, Barcelona, 2008.

[2]

T. G. Elizarova., "Quasi-Gas Dynamic Equations," Computational Fluid and Solid Mechanics, Springer, Dordrecht, 2009. doi: 10.1007/978-3-642-00292-2.

[3]

T. G. Elizarova, Time averaging as an approximate technique for constructing quasi-gas-dynamic and quasi-hydrodynamic equations, Comp. Math. Math. Phys., 51 (2011), 1973-1982. doi: 10.1134/S0965542511110078.

[4]

T. G. Elizarova and S. D. Ustjugov, Quasi-gas dynamics algorithm to solve equations of magnetohydrodynamics. One-dimensional case, (in Russian) Keldysh Inst. Appl. Math. Moscow, preprint No. 1, (2011). Available from: http://www.keldysh.ru/papers/2011/source/prep2011_01.pdf.

[5]

T. G. Elizarova and S. D. Ustjugov, Quasi-gas dynamics algorithm to solve equations of magnetohydrodynamics. Multidimensional case, (in Russian) Keldysh Inst. Appl. Math. Moscow, preprint No. 30, (2011). Available from: http://www.keldysh.ru/papers/2011/source/prep2011_30.pdf.

[6]

A. C. Eringen and G. A. Maugin, "Electrodynamics of Continua II: Fluids and Complex Media," Springer-Verlag, New York, 1990.

[7]

E. Feireisl and A. Novotný, "Singular Limits in Thermodynamics of Viscous Fluids," Advances in Mathematical Fluid Mechanics, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8843-0.

[8]

I. A. Kvasnikov, "Thermodynamics and Statistical Physics, Vol. 1: Theory of Equilibrium Systems and Thermodynamics," (in Russian) Editorial URSS, Moscow, 2002.

[9]

L. D. Landau and E. M. Lifshitz, "Electrodynamics of Continuous Media," Pergamon Press, Oxford, 1960.

[10]

Yu. V. Sheretov, "Continuum Dynamics Under Spatiotemporal Averaging," (in Russian) RKhD, Moscow-Izhevsk, 2009.

[11]

A. A. Zlotnik, Classification of some modifications of the Euler system of equations, Doklady Math., 73 (2006), 302-306. doi: 10.1134/S1064562406020396.

[12]

A. A. Zlotnik, Energy equalities and estimates for barotropic quasi-gas-dynamic and quasi-hydrodynamic systems of equations, Comp. Maths. Math. Phys., 50 (2010), 310-321. doi: 10.1134/S0965542510020120.

[13]

A. A. Zlotnik, A quasi-gas-dynamic system of equations with general state equations, Doklady Math., 81 (2010), 312-316. doi: 10.1134/S1064562410020419.

[14]

A. A. Zlotnik, Linearized stability of equilibrium solutions to the quasi-gas-dynamic system of equations, Doklady Math., 82 (2010), 811-815. doi: 10.1134/S1064562410050352.

[15]

A. A. Zlotnik, On the quasi-gas-dynamic system of equations with general equations of state and a heat source, (in Russian) Math. Modelling, 22 (2010), 53-64.

[16]

A. A. Zlotnik, On construction of quasi-gas-dynamic systems of equations and the barotropic system with the potential body force, (in Russian) Math. Modelling, 24 (2012), 65-79.

[17]

A. A. Zlotnik and B. N. Chetverushkin, Parabolicity of the quasi-gas-dynamic system of equations, its hyperbolic second-order modification, and the stability of small perturbations for them, Comp. Maths. Math. Phys., 48 (2008), 420-446. doi: 10.1134/S0965542508030081.

[18]

A. Zlotnik and V. Gavrilin, On quasi-gas-dynamic system of equations with general equations of state and its application, Math. Modelling Anal., 16 (2011), 509-526. doi: 10.3846/13926292.2011.627382.

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