The kernel is a function of (and of the SPH smoothing length ), i.e., its gradient is given by
Various types of spherically symmetric kernels have been suggested over the years [198, 20]. Among those the spline kernel of Monaghan and 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 it has compact support.
In the Newtonian case is given by [203, 258].
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 . 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 [269, 16, 167, 275]. Steinmetz and 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.
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