where the summation is over all particles other than particle a, p is the pressure, is the density, and d / dt denotes the Lagrangian time derivative. is the artificial viscosity tensor, which is required in SPH to handle shock waves. It poses a major obstacle in extending SPH to relativistic flows (see, e.g., [77, 30]). is the interpolating kernel, and denotes the gradient of the kernel taken with respect to the coordinates of particle a .
The kernel is a function of (and of the SPH smoothing length ), i.e., its gradient is given by
where is a scalar function which is symmetric in a and b, and is a shorthand for . Hence, the forces between particles are along the line of centers.
Various types of spherically symmetric kernels have been suggested over the years [120, 12]. Among those the spline kernel of Monaghan & Lattanzio , mostly used in current SPH-codes, yields the best results. It reproduces constant densities exactly in 1D, if the particles are placed on a regular grid of spacing , and has compact support.
In the Newtonian case is given by 
provided , and otherwise. Here , is the average sound speed, , and is a parameter.
Using the first law of thermodynamics and applying the SPH formalism one can derive the thermal energy equation in terms of the specific internal energy (see, e.g., ). However, when deriving dissipative terms for SPH guided by the terms arising from Riemann solutions, there are advantages to use an equation for the total specific energy , which reads 
where is the artificial energy dissipation term derived by Monaghan . For the relativistic case the explicit form of this term is given in Section 4.2 .
In SPH calculations the density is usually obtained by summing up the individual particle masses, but a continuity equation may be solved instead, which is given by
The capabilities and limits of SPH have been explored, e.g., in [169, 172]. Steinmetz & Müller  conclude that it is possible to handle even difficult hydrodynamic test problems involving interacting strong shocks with SPH provided a sufficiently large number of particles is used in the simulations. SPH and finite volume methods are complementary methods to solve the hydrodynamic equations, each having its own merits and defects.
|Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller
© Max-Planck-Gesellschaft. ISSN 1433-8351
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