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Numerical Study of Singularity Formation in Relativistic Euler Flows

Published online by Cambridge University Press:  03 June 2015

Pierre A. Gremaud  and
Yi Sun
Pierre A. Gremaud*
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA
Yi Sun*
Affiliation:
Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA
*
Email:gremaud@ncsu.edu
Corresponding author.Email:yisun@math.sc.edu
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Abstract

The formation of singularities in relativistic flows is not well understood. Smooth solutions to the relativistic Euler equations are known to have a finite lifespan; the possible breakdown mechanisms are shock formation, violation of the subluminal conditions and mass concentration. We propose a new hybrid Glimm/central-upwind scheme for relativistic flows. The scheme is used to numerically investigate, for a family of problems, which of the above mechanisms is involved.

Keywords

35L6765M0865M2576Y0583C0583C75Relativistic Euler equationssingularity formationGlimm schemecentral-upwind schemehybrid method.
Type
Research Article
Information
Communications in Computational Physics , Volume 16 , Issue 2 , August 2014 , pp. 348 - 364
Copyright
Copyright © Global Science Press Limited 2014

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References

[1]Anile, A. M., Relativistic Fluids and Magnetofluids, Cambridge University Press, London, 1989.Google Scholar
[2]Anninos, P. and Fragile, P., Nonoscillatory central difference and artificial viscosity schemes for relativistic hydrodynamics, Astrophys. J. Suppl. Ser., 144 (2003), 243–257.Google Scholar
[3]Bona, C., Palenzuela-Luque, C., and Bona-Casas, C., Elements of Numerical Relativity and Relativistic Hydrodynamics: From Einstein’s Equations to Astrophysical Simulations, 2nd ed., Lect. Notes in Phys. 783), Springer, Berlin, 2009.Google Scholar
[4]Cannizzo, J. K., Gehrels, N. and Vishniac, E. T., Glimm’s method for relativistic hydrodynamics, Astrophys. J., 680 (2008), 885–896.Google Scholar
[5]Chorin, A. J., Random choice solution of hyperbolic systems, J. Comput. Phys., 22 (1976), 517–533.Google Scholar
[6]Colella, P., Glimm’s method for gas dynamics, SIAM J. Sci. Stat. Comput., 3 (1982), 76–110.Google Scholar
[7]Colella, P. and Woodward, P., The piecewise-parabolic method (PPM) for gas-dynamical simulations, J. Comput. Phys., 54 (1984), 174–201.Google Scholar
[8]Del Zanna, L. and Bucciantini, N., An efficient shock-capturing central-type scheme for multi-dimensional relativistic flows. I. Hydrodynamics, Astron. Astrophys., 390 (2002), 1177–1186.Google Scholar
[9]Dolezal, A. and Wong, S. S. M., Relativistic hydrodynamics and essentially non-oscillatory shock capturing schemes, J. Comput. Phys., 120 (1995), 266–277.Google Scholar
[10]Einfeldt, B., On Godunov-type methods for gas dynamics, SIAM J. Numer. Anal., 25 (1988), 294–318.Google Scholar
[11]Eulderink, F. and Mellema, G., General relativistic hydrodynamics with a Roe solver, Astron. Astrophys. Suppl., 110 (1995), 587–623.Google Scholar
[12]Font, J. A., Numerical hydrodynamics and magnetohydrodynamics in general relativity, Living Rev. Relativity, 11 (2008), http://relativity.livingreviews.org/Articles/lrr-2008-7Google Scholar
[13]Glimm, J., Solutions in the large for nonlinear hyperbolic systems of equations, Commun. Pure Appl. Math., 18 (1965), 697–715.Google Scholar
[14]Godunov, S. K., A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics, Mat. Sb., 47 (1959), 271–290.Google Scholar
[15]Harten, A., Lax, P. D., and van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25 (1983), 35–61.Google Scholar
[16]He, P., Tang, H. Z., An adaptive moving mesh method for two-dimensional relativistic hy-drodynamics, Commun. Comput. Phys., 11 (2012), 114–146.Google Scholar
[17]Hu, J. and Jin, S., On the quasi-random choice method for Liouville equation of geometrical optics with discontinuous wave speed, J. Comput. Math, 31 (2013), 573–591.Google Scholar
[18]Kurganov, A., Noelle, S., and Petrova, G., Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations, SIAM J. Sci. Comput., 23 (2001), 707–740.Google Scholar
[19]Kurganov, A. and Tadmor, E., New high resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys., 160 (2000), 241–282.Google Scholar
[20]Kurganov, A. and Tadmor, E., Solution of two-dimensional Riemann problems for gas-dynamics without Riemann problem solvers, Numer. Methods Partial Differential Equations, 18 (2002), 584–608.Google Scholar
[21]Lax, P., Development of singularity of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys., 5 (1964), 611–613.Google Scholar
[22]LeFloch, P.G. and Yamazaki, M., Entropy solutions of the Euler equations for isothermal relativistic fluids, Int. J. Dynamical Systems and Differential Equations, 1 (2007), 20–37.Google Scholar
[23]LeVeque, R.J., Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, 2002.Google Scholar
[24]Lucas-Serrano, A., Font, J. A., Ibanez, J. M., and Marti, J. M., Assessment of a high-resolution central scheme for the solution of the relativistic hydrodynamic equations, Astron. Astrophys., 428 (2004), 703–715.Google Scholar
[25]Marti, J. M. and Müller, E., The analytical solution of the Riemann problem in relativistic hydrodynamics, J. Fluid Mech., 258 (1994), 317–333.Google Scholar
[26]Marti, J. M. and Müller, E., Extension of the piecewise parabolic method to one-dimensional relativistic hydrodynamics, J. Comput. Phys., 123 (1996), 1–14.Google Scholar
[27]Marti, J. M. and Müller, E., Numerical hydrodynamics in special relativity, Living Rev. Rela-tivity, 6 (2003), http://relativity.livingreviews.org/Articles/lrr-2003-7Google Scholar
[28]Miniati, F., Glimm-Godunov’s method for cosmic-ray-hydrodynamics, J. Comput. Phys., 227 (2007), 776–796.Google Scholar
[29]Pan, R. and Smoller, J.A., Blowup of smooth solutions for relativistic Euler equations, Commun. Math. Phys., 262 (2006), 729–755.Google Scholar
[30]Roe, P. L., Approximate Riemann solvers, parameter vectors and difference schemes, J. Comput. Phys., 43 (1981), 357–372.CrossRefGoogle Scholar
[31]Schneider, V., Katscher, V., Rischke, D. H., Waldhauser, B., Marhun, J. A., and Munz, C.-D., New algorithms for ultra-relativistic numerical hydrodynamics, J. Comput. Phys., 105 (1993), 92–107.Google Scholar
[32]Smoller, J., Shock Waves and Reaction-Diffusion Equations, 2nd ed., Grundlehren Math. Wiss. 258, Springer-Verlag, New York, 1994.Google Scholar
[33]Smoller, J. and Temple, B., Global solutions of the relativistic Euler equations, Commun. Math. Phys., 156 (1993), 6Z-99.Google Scholar
[34]Sod, G., A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys., 22 (19Z8), 1–31.Google Scholar
[35]Taub, A. H., Relativistic fluid mechanics, Annu. Rev. Fluid Mech., 10 (1978), 301–332.Google Scholar
[36]Thompson, K., The special relativistic shock tube. J. Fluid Mech., 171 (1986), 365–375.Google Scholar
[37]Toro, E., Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer, Berlin, 1999.Google Scholar
[38]Wald, R. M., ed. Black holes and Relativistic Stars, University of Chicago Press, Chicago, 1998.Google Scholar
[39]Wen, L., Panaitescu, A., and Laguna, P., A shock-patching code for ultrarelativistic fluid flows, Astrophys. J., 486 (1997), 919–929.Google Scholar
[40]Wilson, J. R. and Mathews, G. J., Relativistic numerical hydrodynamics, Cambridge University Press, Cambridge, 2003.Google Scholar
[41]Yang, J. Y., Chen, M. H., Tsai, I. N., and Chang, J. W., A kinetic beam scheme for relativistic gas dynamics, J. Comput. Phys., 136 (1997), 19–40.Google Scholar
[42]Yang, Z. C., He, P., Tang, H. Z., A direct Eulerian GRP scheme for relativistic hydrodynamics: one-dimensional case, J. Comput. Phys., 230 (2011), 7964–7987.Google Scholar
[43]Yang, Z. C. and Tang, H. Z., A direct Eulerian GRP scheme for relativistic hydrodynamics: two-dimensional case, J. Comput. Phys., 231 (2012), 2116–2139.Google Scholar
[44]Zhang, W. Q. and MacFadyen, A. I., RAM: a relativistic adaptive mesh refinement hydrodynamics code, Astrophys. J. Suppl., 164 (2006), 255–279.Google Scholar
[45]Zhao, J. and Tang, H. Z., RungeKutta discontinuous Galerkin methods with WENO limiter for the special relativistic hydrodynamics, J. Comput. Phys., 242 (2013), 138–168.Google Scholar
[46]Zahran, Y.H., RCM-TVD hybrid scheme for hyperbolic conservation laws, Int. J. Numer. Meth. Fluids, 57 (2007), 745–760.Google Scholar