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Relativistic Smoothed Particle Hydrodynamics

Relativistic Smoothed Particle Hydrodynamics. C.E. Aguiar, T. Kodama U.F. Rio de Janeiro T. Osada,Y. Hama U. São Paulo. Outline Relativistic hydrodynamics Relativistic SPH Entropy-based SPH Shocks and artificial viscosity. Relativistic Hydrodynamics. Energy-momentum conservation.

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Relativistic Smoothed Particle Hydrodynamics

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  1. Relativistic Smoothed Particle Hydrodynamics C.E. Aguiar, T. Kodama U.F. Rio de Janeiro T. Osada,Y. Hama U. São Paulo • Outline • Relativistic hydrodynamics • Relativistic SPH • Entropy-based SPH • Shocks and artificial viscosity

  2. Relativistic Hydrodynamics Energy-momentum conservation Baryon-number conservation

  3. Baryon number conservation: comoving derivative:

  4. enthalpy per baryon: Energy-momentum conservation:

  5. Momentum equation: Energy equation:

  6. Entropy conservation: s = entropy density (rest frame)

  7. Lagrangian Equations

  8. SPH • Developed to study gas dynamics in astrophysical systems. • Lagrangian method. • No grids. • Arbitrary geometries. • Equally applicable in 1, 2 and 3 space • dimensions. - L.Lucy, Astron.J. 82, 1013 (1977) - R.Gingold, J.Monaghan, MNRAS 181, 378 (1977) Reviews: - J. Monaghan, Annu. Rev. Astron. Astrophys. 30, 543 (1992) - L. Hernquist, N. Katz, Ap. J. Suppl. 70, 419 (1989)

  9. Smoothing h x 0 Error:

  10. Particles "Monte-Carlo" sampling nb = baryon number of ''particle'' b

  11. Different ways of writing SP estimates (we omit the SP subscript from now on):

  12. Derivatives No need for finite differences and grids: D D i+1 i-1 i

  13. More than one way of calculating derivatives:

  14. Moving the Particles

  15. Momentum equation Energy equation

  16. Energy and Momentum

  17. Entropy equation

  18. ? Particle Velocity equation for g:

  19. RSPH Equations

  20. Baryon-Free Matter

  21. Lagrangian equations:

  22. Entropy-based RSPH

  23. Ultrarelativistic Pion Gas

  24. Pion Gas Rarefaction Wave

  25. Pion Gas Landau Solution

  26. numerical calculation shock wave x Shock Waves

  27. Pion Gas Shock Wave

  28. Artificial Viscosity

  29. Second Law of Thermodynamics: Thermodynamically normal matter: Thermodynamically anomalous matter:

  30. Dissipative RSPH

  31. Pion Gas Shock Wave

  32. Pion Gas Rankine - Hugoniot:

  33. QGP + Pion Gas

  34. QGP + Pions Rarefaction Shock

  35. QGP + Pions Rarefaction Shock

  36. QGP + Pions Rarefaction Shock

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