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The analytical solution of the Riemann problem in relativistic hydrodynamics

Published online by Cambridge University Press:  26 April 2006

José Ma Martí  and
Ewald Müller
José Ma Martí
Affiliation:
Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 8046 Garching b. München, Germany
Ewald Müller
Affiliation:
Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 8046 Garching b. München, Germany
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Abstract

We consider the decay of an initial discontinuity in a polytropic gas in a Minkowski space–time (the special relativistic Riemann problem). In order to get a general analytical solution for this problem, we analyse the properties of the relativistic flow across shock waves and rarefactions. As in classical hydrodynamics, the solution of the Riemann problem is found by solving an implicit algebraic equation which gives the pressure in the intermediate states. The solution presented here contains as a particular case the special relativistic shock-tube problem in which the gas is initially at rest. Finally, we discuss the impact of this result on the development of high-resolution shock-capturing numerical codes to solve the equations of relativistic hydrodynamics.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 258 , 10 January 1994 , pp. 317 - 333
Copyright
© 1994 Cambridge University Press

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References

Anile, A. M. 1989 Relativistic Fluids and Magnetofluids. Cambridge University Press.
Blandford, R. D. & Mckee, C. F. 1976 Fluid dynamics of relativistic blast waves. Phys. Fluids 19, 1130.Google Scholar
Bogoyavlenski, O. I. 1978 General relativistic self-similar solutions with a spherical shock wave. Soviet Phys. JETP 46, 633.Google Scholar
Centrella, J. M. & Wilson, J. R. 1984 Planar numerical cosmology. II. The difference equations and numerical tests. Astrophys. J. Suppl. Ser. 54, 229.Google Scholar
Courant, R. & Friedrichs, K. O. 1948 Supersonic Flows and Shock Waves. Interscience.
Eltgroth, P. G. 1971 Similarity analysis for relativistic flow in one dimension. Phys. Fluids 14, 2631.Google Scholar
Eltgroth, P. G. 1972 Nonplanar relativistic fluids. Phys. Fluids 15, 2140.Google Scholar
Eulderink, F. & Mellema, G. 1993 Special relativistic jet collimation by inertial confinement. Sterrewacht Leiden Preprint (Astron. Astrophys. submitted).Google Scholar
Font, J. A., Ibáñez, J. MMarquina, A. & Martí, J. M 1993 Multidimensional relativistic hydrodynamics: characteristic fields and modern high-resolution shock-capturing schemes. Astron. Astrophys. (in press).Google Scholar
Godunov, S. K. 1959 A finite difference method for the numerical computation and discontinuous solutions of the equations of fluid dynamics. Mat. Sb. 47, 271.Google Scholar
Israel, W. 1960 Relativistic theory of shock waves. Proc. R. Soc. Lond. A 259, 129.Google Scholar
Johnson, M. H. & Mckee, C. F. 1971 Relativistic hydrodynamics in one dimension. Phys. Rev. D 3, 858.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics, 2nd edn. Pergamon.
Leveque, R. J. 1991 Numerical Methods for Conservation Laws. Birkhäuser.
Liang, E. P. T. 1977 Relativistic simple waves: shock damping and entropy production. Astrophys. J. 211, 361.Google Scholar
Lichnerowicz, A. 1967 Relativistic Hydrodynamics and Magnetohydrodynamics. Benjamin.
Lichnerowicz, A. 1970 Phys. Scripta 2, 221.
Lichnerowicz, A. 1971 Ondes de chocs, ondes infinitesimales et rayons en hydrodynamique et magnetohydrodynamique relativistes. In Relativistic Fluid Dynamics (ed. C. Cattaneo). Cremonese.
Marquina, A., Martí, J. M, Ibántez, J. M, Miralles, J. A. & Donat, R. 1992 Untrarelativistic hydrodynamics: high resolution shock-capturing methods. Astron. Astrophys. 258, 566.Google Scholar
Martí, J. M, Ibáñez, J. M, & Miralles, J. A. 1991 Numerical relativistic hydrodynamics: local characteristic approach. Phys. Rev. D 43, 3794.Google Scholar
Muscato, O. 1988 Breaking of relativistic simple waves. J. Fluid Mech. 196, 223.Google Scholar
Norman, M. L. & Winkler, K.-H. A. 1986 Why ultrarelativistic numerical hydrodynamics is difficult? In Astrophysical Radiation Hydrodynamics (ed. M. L. Norman & K.-H. A. Winkler). Reidel.
Phinney, S. 1987 How fast can a blob go? In Superluminal Radio Sources (ed. J. A. Zensus & T. J. Pearson). Cambridge University Press.
Schneider, V., Katscher, U., Rischke, D. H., Waldhauser, B., Maruhn, J. A. & Munz, C.-D. 1993 New algorithms for ultra-relativistic numerical hydrodynamics. J. Comput. Phys. 105, 92.Google Scholar
Smoller, J. & Temple, B. 1993 Global solutions of the relativistic Euler equations. Commun. Math. Phys. (in press).Google Scholar
Sod, G. A. 1978 A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys. 27, 1.Google Scholar
Strottman, D. 1989 Relativistic hydrodynamics and heavy ion reactions. In Relativistic Fluid Dynamics (ed. A. Anile & Y. Choquet-Bruhat).Springer.
Taub, A. H. 1948 Relativistic Rankine–Hugoniot relations. Phys. Rev. 74, 328.Google Scholar
Taub, A. J. 1978 Relativistic fluid mechanics. Ann. Rev. Fluid Mech. 10, 301.Google Scholar
Thompson, K. W. 1986 The special relativistic shock tube. J. Fluid Mech. 171, 365.Google Scholar
Thorne, K. S. 1973 Relativistic shocks: the Taub adiabat. Astrophys. J. 179, 897.Google Scholar
Woodward, P. R. & Colella, P. 1984 The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54, 115.Google Scholar