×

Kinetic schemes for the ultra-relativistic Euler equations. (English) Zbl 1061.76068

Summary: We present a kinetic numerical scheme for relativistic Euler equations, which describe the flow of a perfect fluid in terms of the particle density \(n\), the spatial part of the four-velocity \(\mathbf u\) and the pressure \(p\). The kinetic approach is very simple in the ultra-relativistic limit, but may also be applied to more general cases. The basic ingredients of the kinetic scheme are the phase-density in equilibrium and the free flight. The phase-density generalizes the non-relativistic Maxwellian for a gas in local equilibrium. The free flight is given by solutions of a collision free kinetic transport equation. The scheme presented here is an explicit method and unconditionally stable. We establish that the conservation laws of mass, momentum and energy as well as the entropy inequality are everywhere exactly satisfied by the solution of the kinetic scheme. For that reason we obtain weak admissible Euler solutions including arbitrarily complicated shock interactions. In the numerical case studies the results obtained from the kinetic scheme are compared with the first order upwind and centered schemes.

MSC:

76M28 Particle methods and lattice-gas methods
76Y05 Quantum hydrodynamics and relativistic hydrodynamics
82C80 Numerical methods of time-dependent statistical mechanics (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Phys. Rev., 94, 511 (1954) · Zbl 0055.23609
[2] Cercignani, C., The Boltzmann equation and its applications, Appl. Math. Sci., 67 (1988) · Zbl 0646.76001
[3] Chernikov, N. A., Equilibrium distribution of the relativistic gas, Acta Phys. Pol., 26, 1069-1092 (1964)
[4] Chernikov, N. A., Microscopic foundation of relativistic hydrodynamics, Acta Phys. Pol., 27, 465-489 (1964) · Zbl 0128.45105
[5] Courant, R.; Friedrichs, K. O., Supersonic Flow and Shock Waves (1948), Interscience Publishers: Interscience Publishers New York · Zbl 0041.11302
[6] C.M. Dafermos, Entropy and the stability of classical solutions of hyperbolic systems of conservation laws, in: T. Ruggeri (Ed.), Recent Mathematical Methods in Nonlinear Wave Propagation, Montecatini terme, 1994, pp. 48-69; C.M. Dafermos, Entropy and the stability of classical solutions of hyperbolic systems of conservation laws, in: T. Ruggeri (Ed.), Recent Mathematical Methods in Nonlinear Wave Propagation, Montecatini terme, 1994, pp. 48-69 · Zbl 0878.35072
[7] S.M. Deshpande, R. Raul, Kinetic theory based fluid-in-cell method for Eulerian fluid dynamics, Rep. 82 FM 14, Department of Aerospace Engineering, Indian Institute of Science (Bangalore, India), Huly, 1982; S.M. Deshpande, R. Raul, Kinetic theory based fluid-in-cell method for Eulerian fluid dynamics, Rep. 82 FM 14, Department of Aerospace Engineering, Indian Institute of Science (Bangalore, India), Huly, 1982
[8] S.M. Deshpande, A second order accurate, kinetic-theory based, method for inviscid compressible flows, NASA Langley Tech. paper No. 2613, 1986; S.M. Deshpande, A second order accurate, kinetic-theory based, method for inviscid compressible flows, NASA Langley Tech. paper No. 2613, 1986
[9] Deshpande, S. M.; Kulkarni, P. S., New developments in kinetic schemes, Comput. Math. Appl., 35, 75-93 (1998) · Zbl 0911.76054
[10] Deshpande, S. M., Kinetic flux splitting schemes, (Hafez, M.; Oshima, K., Computational Fluid Dynamics Review 1995 (1995), Wiley: Wiley New York) · Zbl 0875.76360
[11] Dreyer, W.; Kunik, M., The maximum entropy principle revisited, Cont. Mech. Thermodyn., 10, 331-347 (1998) · Zbl 0951.76085
[12] Dreyer, W.; Kunik, M., Initial and boundary value problems of hyperbolic heat conduction, Cont. Mech. Thermodyn., 11.4, 227-245 (1999) · Zbl 0931.35175
[13] Eckart, C., The thermodynamics of irreversible process I: the simple fluid, Phys. Rev., 58, 267-269 (1940) · JFM 66.1076.03
[14] Eckart, C., The thermodynamics of irreversible process II: fluid mixtures, Phys. Rev., 58, 269-275 (1940) · JFM 66.1077.01
[15] Eckart, C., The thermodynamics of irreversible process III: relativistic theory of the simple fluid, Phys. Rev., 58, 919-928 (1940) · JFM 66.1077.02
[16] Friedrichs, K. O.; Lax, P. D., Systems of conservation equations with a convex extension, Proc. Acad. Sci. USA, 68, 1686 (1971) · Zbl 0229.35061
[17] Godlewski, E.; Raviart, P. A., Numerical approximation of hyperbolic systems of conservation laws, Appl. Math. Sci., 118 (1996) · Zbl 0860.65075
[18] deGroot, S. R.; van Leeuven, W. A.; van Weert, Ch. G., Relativistic Kinetic Theory. Principles and Applications (1980), North Holland: North Holland Amsterdam
[19] Israel, W., Nonstationary irreversible thermodynamics: a causal relativistic theory, Ann. Phys. (NY), 100, 310-331 (1976)
[20] Jüttner, F., Das Maxwellsche Gesetz der Geschwindigkeitsverteilung in der relativtheorie, Ann. Phys. (Leipzig), 34, 856-882 (1911) · JFM 42.0981.01
[21] Junk, M., A new perspective on kinetic schemes, SIAM Journal on Numerical Analysis, 38, 1603-1625 (2000) · Zbl 0982.35065
[22] Jüttner, F., Die Relativistische Quantentheorie des idealen gases, Z. Phys., 47, 542-566 (1928) · JFM 54.0987.01
[23] Martí, J. M.; Müller, E., Numerical hydrodynamics in special relativity, Living Rev. Relativity, 2, 1-101 (1999)
[24] Oleinik, O. A., Discontinuous solutions of nonlinear differential equations, Am. Math. Soc. Trans. Serv., 26, 95-172 (1957) · Zbl 0131.31803
[25] Reitz, R. D., One-dimensional compressible gas dynamics calculations using the Boltzmann equation, J. Comput. Phys., 42, 1, 108-123 (1981) · Zbl 0466.76069
[26] Weinberg, S., Gravitation and Cosmology (1972), Wiley: Wiley New York
[27] Xu, K., BGK-based scheme for multicomponent flow calculations, J. Comput. Phys., 134, 122-133 (1997) · Zbl 0882.76060
[28] K. Xu, Gas kinetic schemes for unsteady compressible flow simulations, 29th CFD, Lecture Series, 1998; K. Xu, Gas kinetic schemes for unsteady compressible flow simulations, 29th CFD, Lecture Series, 1998
[29] K. Xu, Gas-kinetic theory based flux slitting method for ideal magnetohydrodynamics, ICASE Report No. Tr. 98-53, 1998; K. Xu, Gas-kinetic theory based flux slitting method for ideal magnetohydrodynamics, ICASE Report No. Tr. 98-53, 1998
[30] K. Xu, Gas evolution dynamics in Godunov-type schemes and analysis of numerical shock instability, ICASE Report No. Tr. 99-6, 1998; K. Xu, Gas evolution dynamics in Godunov-type schemes and analysis of numerical shock instability, ICASE Report No. Tr. 99-6, 1998
[31] Yang, J. Y.; Chen, M. H.; Tsai, I. N.; Chang, J. W., A kinetic beam scheme for relativistic gas dynamics, J. Comput. Phys., 136, 19-40 (1997) · Zbl 0889.76053
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.