### 9.6 Newtonian SPH equations

Following Monaghan [200], the SPH equation of motion for a particle a with mass and velocity
is given by
where the summation is over all particles other than particle a, is the pressure, is the density, and
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.,
[130, 53]). 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 and , 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 [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]

Here , is the average sound speed, , and is
a parameter. Alternative forms of the artificial viscosity are discussed in [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., [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]

where is the artificial energy dissipation term derived by Monaghan [200]. 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 [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.