A comprehensive discussion of SPH can be found in
the reviews of Hernquist and Katz [124], Benz [20], and
Monaghan [198
, 199
]. The
non-relativistic SPH equations are briefly discussed in
Section 9.6. The capabilities and limits of
SPH are explored, e.g., in [269
, 16
, 167
, 275
], and the stability
of the SPH algorithm is investigated in [271].
The SPH equations for special relativistic flows
have been first formulated by Monaghan [198]. Monaghan and
Price [202
] showed how the
equations of motion for the SPH method may be derived from a
variational principle for both non-relativistic and (special and
general) relativistic flows when there is no dissipation. For
relativistic flows the SPH equations given in Section 9.6 can be used except that each SPH
particle
carries
baryons instead of mass
[198
, 53
]. Hence, the rest
mass of particle
is given by
, where
is the baryon rest mass (if the fluid is made of
baryons). Transforming the notation used in [53
] to ours, the
continuity equation, the momentum, and the total energy equations
for particle
are given by (unit of velocity is
)
Special relativistic flow problems have been
simulated with SPH by [151, 134
, 172
, 174, 53
, 261
]. Extensions of SPH
capable of treating general relativistic flows have been considered
by [134, 150
, 261
, 202
, 204
]. Concerning
relativistic SPH codes the artificial viscosity is the most
critical issue. It is required to handle shock waves properly, and
ideally it should be predicted by a relativistic kinetic theory for
the fluid. However, unlike its Newtonian analogue, the relativistic
theory has not yet been developed to the degree required to achieve
this.
For Newtonian SPH, Lattanzio et al. [155] have shown that a
viscosity quadratic in the velocity divergence is necessary in high
Mach number flows. They proposed a form such that the viscous
pressure could be simply added to the fluid pressure in the
equation of motion and the energy equation. As this simple form of
the artificial viscosity has known limitations, they also proposed
a more sophisticated form of the artificial viscosity terms, which
leads to a modified equation of motion. This artificial viscosity
works much better, but it cannot be generalized to the relativistic
case in a consistent way. Utilizing an equation for the specific
internal energy, both Mann [172] and Laguna et
al. [150
] use such an
inconsistent formulation. Their artificial viscosity term is not
included in the expression of the specific relativistic enthalpy.
In a second approach, Mann [172
] allows for a
time-dependent smoothing length and SPH particle mass, and further
proposes an SPH variant based on the total energy equation.
Lahy [151] and Siegler and
Riffert [261
] use a consistent
artificial viscosity pressure added to the fluid pressure. Siegler
and Riffert [261
] have also
formulated the hydrodynamic equations in conservation form (see
also [202
]).
Monaghan [200] incorporates
concepts from Riemann solvers into SPH (see also [129]). For this reason he
also proposes to use a total energy equation in SPH simulation
instead of the commonly used internal energy equation, which would
involve time derivatives of the Lorentz factor in the relativistic
case. Chow and Monaghan [53
] have extended this
concept and have proposed an SPH algorithm, which gives good
results when simulating an ultra-relativistic gas. In both cases
the intention was not to introduce Riemann solvers into the SPH
algorithm, but to use them as a guide to improve the artificial
viscosity required in SPH. Multi-dimensional simulations of general
relativistic flows (in a given time-independent metric) using the
SPH formulation of Monaghan and Price [202
] and the SPH
algorithm of Chow and Monaghan [53
] have been performed
by Muir [204
].
In Roe’s Riemann solver [247], as well as in its
relativistic variant proposed by Eulerdink [83
, 84
] (see Section
3.4), the numerical flux is computed
by solving a locally linear system, and depends on both the
eigenvalues and (left and right) eigenvectors of the Jacobian
matrix associated to the fluxes and on the jumps in the conserved
physical variables (see Equations (36
) and (37
)). Monaghan [200
] realized that an
appropriate form of the dissipative terms
and
for the interaction between
particles
and
can be obtained by treating the
particles as the equivalent of left and right states taken with
reference to the line joining the particles. The quantity
corresponding to the eigenvalues (wave propagation speeds) is an
appropriate signal velocity
(see below), and that
equivalent to the jump across characteristics is a jump in the
relevant physical variable. For the artificial viscosity tensor,
, Monaghan [200
] assumes that the
jump in velocity across characteristics can be replaced by the
velocity difference between
and
along the line
joining them.
With these considerations in mind, Chow and
Monaghan [53] proposed for
in the relativistic case the form
The dissipation term in the energy equation is
derived in a similar way, and is given by [53]
To determine the signal velocity, Chow and
Monaghan [53] (and
Monaghan [200
] in the
non-relativistic case) start from the (local) eigenvalues, and
hence the wave velocities
and
of one-dimensional relativistic hydrodynamic flows.
Again considering particles
and
as the left and right
states of a Riemann problem with respect to motions along the line
joining the particles, the appropriate signal velocity is the speed
of approach (as seen in the computing frame) of the signal sent
from
towards
and that from
to
. This is the natural speed for the sharing of
physical quantities, because when information about the two states
meets it is time to construct a new state. This speed of approach
should be used when determining the size of the time step by the
Courant condition (for further details see [53
]).
Chow and Monaghan [53] have demonstrated
the performance of their Riemann problem guided relativistic SPH
algorithm by calculating several shock tube problems involving
ultra-relativistic speeds up to
. The
algorithm gives good results, but finite volume schemes based on
Riemann solvers give more accurate results and can handle even
larger speeds (see Section 6).