Skip to main content Accessibility help
×
Home

Numerical Study of Singularity Formation in Relativistic Euler Flows

  • Pierre A. Gremaud (a1) and Yi Sun (a2)

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.

Copyright

Corresponding author

Corresponding author.Email:yisun@math.sc.edu

References

Hide All
[1]Anile, A. M., Relativistic Fluids and Magnetofluids, Cambridge University Press, London, 1989.
[2]Anninos, P. and Fragile, P., Nonoscillatory central difference and artificial viscosity schemes for relativistic hydrodynamics, Astrophys. J. Suppl. Ser., 144 (2003), 243257.
[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.
[4]Cannizzo, J. K., Gehrels, N. and Vishniac, E. T., Glimm’s method for relativistic hydrodynamics, Astrophys. J., 680 (2008), 885896.
[5]Chorin, A. J., Random choice solution of hyperbolic systems, J. Comput. Phys., 22 (1976), 517533.
[6]Colella, P., Glimm’s method for gas dynamics, SIAM J. Sci. Stat. Comput., 3 (1982), 76110.
[7]Colella, P. and Woodward, P., The piecewise-parabolic method (PPM) for gas-dynamical simulations, J. Comput. Phys., 54 (1984), 174201.
[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), 11771186.
[9]Dolezal, A. and Wong, S. S. M., Relativistic hydrodynamics and essentially non-oscillatory shock capturing schemes, J. Comput. Phys., 120 (1995), 266277.
[10]Einfeldt, B., On Godunov-type methods for gas dynamics, SIAM J. Numer. Anal., 25 (1988), 294318.
[11]Eulderink, F. and Mellema, G., General relativistic hydrodynamics with a Roe solver, Astron. Astrophys. Suppl., 110 (1995), 587623.
[12]Font, J. A., Numerical hydrodynamics and magnetohydrodynamics in general relativity, Living Rev. Relativity, 11 (2008), http://relativity.livingreviews.org/Articles/lrr-2008-7
[13]Glimm, J., Solutions in the large for nonlinear hyperbolic systems of equations, Commun. Pure Appl. Math., 18 (1965), 697715.
[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), 271290.
[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), 3561.
[16]He, P., Tang, H. Z., An adaptive moving mesh method for two-dimensional relativistic hy-drodynamics, Commun. Comput. Phys., 11 (2012), 114146.
[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), 573591.
[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), 707740.
[19]Kurganov, A. and Tadmor, E., New high resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys., 160 (2000), 241282.
[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), 584608.
[21]Lax, P., Development of singularity of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys., 5 (1964), 611613.
[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), 2037.
[23]LeVeque, R.J., Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, 2002.
[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), 703715.
[25]Marti, J. M. and Müller, E., The analytical solution of the Riemann problem in relativistic hydrodynamics, J. Fluid Mech., 258 (1994), 317333.
[26]Marti, J. M. and Müller, E., Extension of the piecewise parabolic method to one-dimensional relativistic hydrodynamics, J. Comput. Phys., 123 (1996), 114.
[27]Marti, J. M. and Müller, E., Numerical hydrodynamics in special relativity, Living Rev. Rela-tivity, 6 (2003), http://relativity.livingreviews.org/Articles/lrr-2003-7
[28]Miniati, F., Glimm-Godunov’s method for cosmic-ray-hydrodynamics, J. Comput. Phys., 227 (2007), 776796.
[29]Pan, R. and Smoller, J.A., Blowup of smooth solutions for relativistic Euler equations, Commun. Math. Phys., 262 (2006), 729755.
[30]Roe, P. L., Approximate Riemann solvers, parameter vectors and difference schemes, J. Comput. Phys., 43 (1981), 357372.
[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), 92107.
[32]Smoller, J., Shock Waves and Reaction-Diffusion Equations, 2nd ed., Grundlehren Math. Wiss. 258, Springer-Verlag, New York, 1994.
[33]Smoller, J. and Temple, B., Global solutions of the relativistic Euler equations, Commun. Math. Phys., 156 (1993), 6Z-99.
[34]Sod, G., A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys., 22 (19Z8), 131.
[35]Taub, A. H., Relativistic fluid mechanics, Annu. Rev. Fluid Mech., 10 (1978), 301332.
[36]Thompson, K., The special relativistic shock tube. J. Fluid Mech., 171 (1986), 365375.
[37]Toro, E., Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer, Berlin, 1999.
[38]Wald, R. M., ed. Black holes and Relativistic Stars, University of Chicago Press, Chicago, 1998.
[39]Wen, L., Panaitescu, A., and Laguna, P., A shock-patching code for ultrarelativistic fluid flows, Astrophys. J., 486 (1997), 919929.
[40]Wilson, J. R. and Mathews, G. J., Relativistic numerical hydrodynamics, Cambridge University Press, Cambridge, 2003.
[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), 1940.
[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), 79647987.
[43]Yang, Z. C. and Tang, H. Z., A direct Eulerian GRP scheme for relativistic hydrodynamics: two-dimensional case, J. Comput. Phys., 231 (2012), 21162139.
[44]Zhang, W. Q. and MacFadyen, A. I., RAM: a relativistic adaptive mesh refinement hydrodynamics code, Astrophys. J. Suppl., 164 (2006), 255279.
[45]Zhao, J. and Tang, H. Z., RungeKutta discontinuous Galerkin methods with WENO limiter for the special relativistic hydrodynamics, J. Comput. Phys., 242 (2013), 138168.
[46]Zahran, Y.H., RCM-TVD hybrid scheme for hyperbolic conservation laws, Int. J. Numer. Meth. Fluids, 57 (2007), 745760.

Keywords

Related content

Powered by UNSILO

Numerical Study of Singularity Formation in Relativistic Euler Flows

  • Pierre A. Gremaud (a1) and Yi Sun (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.