2.2 SRHD as a hyperbolic 2 Special Relativistic Hydrodynamics2 Special Relativistic Hydrodynamics

2.1 Equations 

Using the Einstein summation convention, the equations describing the motion of a relativistic fluid are given by the five conservation laws



where tex2html_wrap_inline5673, and where denotes the covariant derivative with respect to coordinate tex2html_wrap_inline5677 . Furthermore, tex2html_wrap_inline5637 is the proper rest-mass density of the fluid, tex2html_wrap_inline5681 its four-velocity, and tex2html_wrap_inline5683 is the stress-energy tensor, which for a perfect fluid can be written as


Here tex2html_wrap_inline5685 is the metric tensor, p the fluid pressure, and and h the specific enthalpy of the fluid defined by


where tex2html_wrap_inline5691 is the specific internal energy. Note that we use natural units (i.e., the speed of light c =1) throughout this review.

In Minkowski spacetime and Cartesian coordinates tex2html_wrap_inline5695, the conservation equations (1Popup Equation2Popup Equation) can be written in vector form as


where i = 1,2,3. The state vector tex2html_wrap_inline5699 is defined by


and the flux vectors tex2html_wrap_inline5701 are given by


The five conserved quantities D, tex2html_wrap_inline5705, tex2html_wrap_inline5707, tex2html_wrap_inline5709 and tex2html_wrap_inline5711 are the rest-mass density, the three components of the momentum density, and the energy density (measured relative to the rest mass energy density), respectively. They are all measured in the laboratory frame, and are related to quantities in the local rest frame of the fluid (primitive variables) through




where tex2html_wrap_inline5713 are the components of the three-velocity of the fluid


and W is the Lorentz factor


The system of equations (5Popup Equation) with definitions (6Popup Equation, 8Popup Equation, 9Popup Equation, 10Popup Equation, 11Popup Equation, 12Popup Equation) is closed by means of an equation of state (EOS), which we shall assume to be given in the form


In the non-relativistic limit (i.e., tex2html_wrap_inline5717, tex2html_wrap_inline5719) D, tex2html_wrap_inline5723 and tex2html_wrap_inline5711 approach their Newtonian counterparts tex2html_wrap_inline5637, tex2html_wrap_inline5729 and tex2html_wrap_inline5731, and the equations of system (5Popup Equation) reduce to the classical ones. In the relativistic case the equations of (5Popup Equation) are strongly coupled via the Lorentz factor and the specific enthalpy, which gives rise to numerical complications (see Section  2.3).

In classical numerical hydrodynamics it is very easy to obtain tex2html_wrap_inline5713 from the conserved quantities (i.e., tex2html_wrap_inline5637 and tex2html_wrap_inline5729). In the relativistic case, however, the task to recover tex2html_wrap_inline5739 from tex2html_wrap_inline5741 is much more difficult. Moreover, as state-of-the-art SRHD codes are based on conservative schemes where the conserved quantities are advanced in time, it is necessary to compute the primitive variables from the conserved ones one (or even several) times per numerical cell and time step making this procedure a crucial ingredient of any algorithm (see Section  9.1).

2.2 SRHD as a hyperbolic 2 Special Relativistic Hydrodynamics2 Special Relativistic Hydrodynamics

image Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de