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. Energy-momentum conservation:
enthalpy per baryon:

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
Error:

10. nb = baryon number of ''particle'' b
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
i+1

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. Shock Waves
shock wave x
numerical calculation

27. Pion Gas
Shock Wave

28. Artificial Viscosity

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

31. Dissipative RSPH

32. Pion Gas
Shock Wave

33. Pion Gas
Rankine - Hugoniot:

34. QGP + Pion Gas

35. QGP + Pions
Rarefaction Shock

36. QGP + Pions
Rarefaction Shock

37. QGP + Pions
Rarefaction Shock