Relativistic Smoothed Particle Hydrodynamics C E Aguiar T

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 Baryon-number conservation

Baryon number conservation: comoving derivative:

Energy-momentum conservation: enthalpy per baryon:

Momentum equation: Energy equation:

Entropy conservation: s = entropy density (rest frame)

Lagrangian Equations

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)

Smoothing h 0 Error: x

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

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

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

More than one way of calculating derivatives:

Moving the Particles

Momentum equation Energy equation

Energy and Momentum

Entropy equation

Particle Velocity ? equation for g:

RSPH Equations

Baryon-Free Matter

Lagrangian equations:

Entropy-based RSPH

Ultrarelativistic Pion Gas

Pion Gas Rarefaction Wave

Pion Gas Landau Solution

Shock Waves numerical calculation shock wave x

Pion Gas Shock Wave

Artificial Viscosity

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

Dissipative RSPH

Pion Gas Shock Wave

Pion Gas Rankine - Hugoniot:

QGP + Pion Gas

QGP + Pions Rarefaction Shock

QGP + Pions Rarefaction Shock

QGP + Pions Rarefaction Shock