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 [123], 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 [122
]
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., [121]). 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 [122
]
where
is the artificial energy dissipation term derived by
Monaghan [122]. 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 [169] 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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Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller http://www.livingreviews.org/lrr-1999-3 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |