World Academy of Science, Engineering and Technology
Published January 1, 2015 | Version 10000315
Journal article Open

Central Finite Volume Methods Applied in Relativistic Magnetohydrodynamics: Applications in Disks and Jets

Description

We have developed a new computer program in
Fortran 90, in order to obtain numerical solutions of a system
of Relativistic Magnetohydrodynamics partial differential equations
with predetermined gravitation (GRMHD), capable of simulating
the formation of relativistic jets from the accretion disk of matter
up to his ejection. Initially we carried out a study on numerical
methods of unidimensional Finite Volume, namely Lax-Friedrichs,
Lax-Wendroff, Nessyahu-Tadmor method and Godunov methods
dependent on Riemann problems, applied to equations Euler in
order to verify their main features and make comparisons among
those methods. It was then implemented the method of Finite
Volume Centered of Nessyahu-Tadmor, a numerical schemes that
has a formulation free and without dimensional separation of
Riemann problem solvers, even in two or more spatial dimensions,
at this point, already applied in equations GRMHD. Finally, the
Nessyahu-Tadmor method was possible to obtain stable numerical
solutions - without spurious oscillations or excessive dissipation -
from the magnetized accretion disk process in rotation with respect
to a central black hole (BH) Schwarzschild and immersed in a
magnetosphere, for the ejection of matter in the form of jet over a
distance of fourteen times the radius of the BH, a record in terms
of astrophysical simulation of this kind. Also in our simulations,
we managed to get substructures jets. A great advantage obtained
was that, with the our code, we got simulate GRMHD equations in
a simple personal computer.

Files

10000315.pdf

Files (617.4 kB)

Name Size
md5:4d4021d30d00744c460afe650d14848e
617.4 kB Preview Download

Additional details

References

  • P. Anninos, P. C. Fragile, J. D. Salmonson, Cosmos++ : Relativistic Magnetohydrodynamics on Unstructured Grids with Local Adaptive Refinement, Astrophys. J., 635 (2005) 7-23.
  • J. Balbás, E. Tadmor, C.-C. Wu,Non-oscillatory central schemes for one- and two-dimensional MHD equations: I, J. Comput. Phys., 201 (2004) pp. 261-285.
  • V. S. Beskin, Magnetohydrodynamic models of astrophysical jets. Phys. Uspekhi, 53 (2010) 1199-1233.
  • V. S. Beskin, MHD flows in compact astrophysical objects, Springer, Heidelberg, 2010.
  • COCONUT/COCOA.
  • L. Del Zanna, et al., ECHO: an Eulerian Conservative High Order scheme for general relativistic magnetohydrodynamics and magnetodynamics, Astron. Astrophys., 473 (2007) 11-30.
  • S. S. Doeleman, et al., Jet-Launching Structure Resolved Near the Supermassive Black Hole in M87, Science, 338 (2012) 255-258.
  • H. Falcke, F. W. Hehl, The Galactic Black Hole - Lectures on general relativity and astrophysics, IOP, London, 2003.
  • J. A. Font, Numerical Hydrodynamics and Magnetohydrodynamics in General Relativity, Living Rev. Relativity, 11 (2008). [10] C. F. Gammie, J. C. McKinney, G. Toth, HARM: A numerical scheme for general relativistic magnetohydrodynamics, Astrophys. J., 589 (2003) 444–457. [11] R. O. Garcia, Métodos de Volumes Finitos Centrados unsplitting utilizados na obtenção de soluções em Magnetohidrodinâmica: aplicações em discos e jatos, thesis IMECC/UNICAMP, 2014. [12] B. Giacomazzo, L. Rezzolla, WhiskyMHD: a new numerical code for general relativistic magnetohydrodynamics, Class. Quantum Grav., 24 (2007) S235-S255, 2007. [13] S. K. Godunov, A Finite Difference Method for the Computation of Discontinuous Solutions of the Equations of Fluid Dynamics, Mat. Sb., 47 (1959) 357-393. [14] A. Harten, P. D. Lax, B. van Leer, On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws, SIAM Review, 25 (1983) 35-61. 1983. [15] W. Hundsdorfer, J. G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Rection Equations, Springer, New York, 2003. [16] S. Koide, K. Shibata, T. Kudoh, Relativistic Jet Formation from Black Hole Magnetized Accretion Disk: Method, Test and Applications of a General Relativistic Magnetohydrodynamic Numerical Code, Astrophys. J., 522 (1999) 727-752. [17] S. Koide, General relativistic plasmas around rotating black holes. Procedings IAU Symposium, 275 (2011). [18] S. S. Komissarov, et al., Magnetic acceleration of ultra-relativistic jets in gamma-ray burst sources, Mon. Not. R. Astron. Soc., 394 (2009) 1182-1212. [19] R. J. Leveque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, New York, 2002. [20] J. C. McKinney, R. D. Blandford, Stability of relativistic jets from rotating, accreting black holes via fully three-dimensional magnetohydrodynamic simulations, Mon. Not. R. Astron. Soc., 394 (2009) L126-L30. [21] A. Mignone, J. C. McKinney, Equation of state in relativistic magnetohydrodynamics: variable versus constant adiabatic index, Mon. Not. R. Astron. Soc., 378 (2007) 1118-1130. [22] P. Mösta, et al., GRHydro: A new open source general-relativistic magnetohydrodynamics code for the Einstein Toolkit, Class. Quantum Grav., 31 (2014). [23] H. Nessyahu, E. Tadmor, Non-Oscillatory Central Differencing for Hyperbolic Conservation Laws, J. Comput. Phys., 87 (1990) 408-463. [24] K. -I. Nishikawa, et al., A General Relativistic Magnetohydrodynamic Simulation of Jet Formation, Astrophys. J., 625 (2005) 60-71. [25] W. H. Press, et al., Numerical Recipes in Fortran 90. Cambridge University Press, Second Edtion, United States of American, 1997. [26] M. J. D. Powell, Approximation theory and methods, Cambridge University Press, Cambridge, 2001. [27] V. Schneider, et al., New Algorithms for Ultra-relativistic Numerical Hydrodynamics, J. Comp. Phys., vol. 105, n. 1, pp. 92-107, 1993. [28] S. L. Shapiro, S. A. Teukolsky, Black Holes, White Dwarfs and Neutron Stars - The Physics of Compact Objects, John Wiley & Sons, New York, 1983. [29] A. Tchekhovskoy, J. C. McKinney, R. Narayan, WHAM: a WENO-based general relativistic numerical scheme – I. Hydrodynamics, Mon. Not. R. Astron. Soc., 379 (2007) 469–497. [30] J. W. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, Springer, New York, 1995. [31] K. S. Thorne, R. H. Price, D. A. MacDonald, Black Holes: The Membrane Paradigm, Yale University Press, New Haven, 1986. [32] E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics - A Practical Introduction, third edition, Springer, Germany, 2009. [33] R. M. Wald, Black Hole in an uniform magnetic field, Phys. Rev. D, 10 (1974) 16-80. [34] P. Woodward, P. Colella, The Numerical Simulation of Two-Dimensional Fluid Flow with Strong Shocks, J. Comput. Phys., 54 (1984) 115-173.