1. Smoothed Particle Hydrodynamics
Carlos Eduardo Aguiar
Instituto de Física - UFRJ
Outline
Non-relativistic hydrodynamics.
SPH equations.
Applications.
Relativistic hydrodynamics.
Relativistic SPH.
High energy nuclear physics.

2. Fluid Dynamics

3. Hydrodynamic Equations
Continuity equation
Ideal fluid
Euler’s equation

4. Entropy Equation
no viscosity
no thermal conduction

5. Energy Equation

6. Ideal Gas

7. Conservation Laws

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. Particles
"Monte-Carlo" sampling

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:
Exact Galilean invariance

14. Moving the Particles

15. Euler's Equation
Exact momentum conservation

16. Entropy

17. Energy

18. Alternatively:

19. SPH Equations

20. Smoothing Kernels
Gaussian:
Spline:

21. Shock Waves
shock wave x
numerical calculation

22. Artificial Viscosity Galilean invariant. Vanishes for rotations.
Conserves linear and angular momentum.

23. Local Resolution Length

24. SPH Simulation of the Hubble Volume
Mass density in a thin slice (100x100x20Mpc/h)
at the present epoch. This is a view one would
observe if the speed of light were infinite.

25. SPH Simulation of Galaxy Formation
Gas
Dark Matter
Density of gas and dark matter
in a group of galaxies.

26. SPH Simulation of Supernova Explosion
75 ms after bounce
z (km)
x (km)
Herant et al., Ap.J. 435, 339 (1994)

27. SPH Simulation of Supernova Explosion
Herant et al., Ap.J. 435, 339 (1994)

28. SPH Simulation of Stellar Collision
Disruption of a main sequence star by a close
encounter with a high velocity neutron star.
Colors represent log (density).

29. SPH Simulation of Colliding Asteroids
An 8 m radius rock strikes the 1.6 km long
asteroid Castalia at 5 km/s. Red is totally
fractured rock, blue is intermediate fractured
rock, and white particles represent the impactor.

30. SPH Simulation of Projectile Impact on Sand
Projectile: Aluminum cylinder 30x10 cm (2d).
Initial velocity: 3 km/sec
Particles
Temperature

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

32. Continuity equation: Entropy equation: s = entropy density
(rest frame)

33. Relativistic Euler equation:
w = enthalpy per baryon
Momentum equation:

34. Energy equation:

35. Relativistic SPH

36. Momentum (per baryon number)
Energy (per baryon number)

37. Particle Velocity ?

38. RSPH Equations

39. Baryon-Free System
entropy density:
(Rest frame)
(Lab frame)

40. Baryon-Free RSPH

41. Ultrarelativistic Pion Gas

42. Ultrarelativistic Pions
Rarefaction Wave

43. SHASTA et al.

44. Ultrarelativistic Pions
Landau Solution

45. Artificial Viscosity E.Chow and J.Monaghan, Journal of
Computational Physics 134, 296 (1997)
See also:
S.Siegler and H.Riffert, astro-ph/

46. Shock Tube
Ideal nucleon gas

47. SHASTA et al.

48. Shock Tube
Ideal nucleon gas

49. SHASTA et al.