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 [201], 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 [200
]
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., [199]). 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 [200
]
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 [269] 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.