A comprehensive discussion of SPH can be found in the reviews of Hernquist & Katz [76], Benz [12] and Monaghan [120, 121]. The non-relativistic SPH equations are briefly discussed in Section 9.5 . The capabilities and limits of SPH are explored, e.g., in [169, 172], and the stability of the SPH algorithm is investigated in [170].

The SPH equations for special relativistic flows have been
first formulated by Monaghan [120]. For such flows the SPH equations given in Section
9.5
can be taken over except that each SPH particle
*a*
carries
baryons instead of mass
[120,
30]. Hence, the rest mass of particle
*a*
is given by
, where
is the baryon rest mass (if the fluid is made of baryons).
Transforming the notation used in [30] to ours, the continuity equation, the momentum and the total
energy equations for particle
*a*
are given by (unit of velocity is
*c*)

and

respectively. Here, the summation is over all particles other
than particle
*a*, and
*d*
/
*dt*
denotes the Lagrangian time derivative.

is the baryon number density,

the momentum per particle, and

the total energy per particle (all measured in the laboratory
frame). The momentum density
, the energy density
(measured in units of the rest mass energy density), and the
specific enthalpy
*h*
are defined in Section
2.1
.
and
are the SPH dissipation terms, and
denotes the gradient of the kernel
(see Section
9.5
for more details).

Special relativistic flow problems have been simulated with SPH by [90, 80, 102, 104, 30, 164]. Extensions of SPH capable of treating general relativistic flows have been considered by [80, 89, 164]. 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. [94] have shown that in high Mach number flows a viscosity quadratic in the velocity divergence is necessary. They proposed a form of the artificial viscosity such that the viscous pressure could be simply added to the fluid pressure in the equation of motion and the energy equation. Because 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 [102] and Laguna et al. [89] use such an inconsistent formulation. Their artificial viscosity term is not included into the expression of the specific relativistic enthalpy. In a second approach, Mann [102] allows for a time-dependent smoothing length and SPH particle mass, and further proposed a SPH variant based on the total energy equation. Lahy [90] and Siegler & Riffert [164] use a consistent artificial viscosity pressure added to the fluid pressure. Siegler & Riffert [164] have also formulated the hydrodynamic equations in conservation form.

Monaghan [122] incorporates concepts from Riemann solvers into SPH. For this reason he also proposes to use a total energy equation in SPH simulations instead of the commonly used internal energy equation, which would involve time derivatives of the Lorentz factor in the relativistic case. Chow & Monaghan [30] 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.

In Roe's Riemann solver [155], as well as in its relativistic variant proposed by
Eulerdink [49,
50] (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 Eqs. (26) and (27)). Monaghan [122] realized that an appropriate form of the dissipative terms
and
for the interaction between particles
*a*
and
*b*
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 [122] assumes that the jump in velocity across characteristics can be
replaced by the velocity difference between
*a*
and
*b*
along the line joining them.

With these considerations in mind Chow & Monaghan [30] proposed for in the relativistic case the form

when particles
*a*
and
*b*
are approaching, and
otherwise. Here
*K*
= 0.5 is a dimensionless parameter, which is chosen to have the
same value as in the non-relativistic case [122].
is the average baryon number density, which has to be present
in (53), because the pressure terms in the summation of (90) have an extra density in the denominator arising from the SPH
interpolation. Furthermore,

is the unit vector from
*b*
to
*a*, and

where

Using instead of (see Eq. (51)) the modified momentum , which involves the line of sight velocity , guarantees that the viscous dissipation is positive definite [30].

The dissipation term in the energy equation is derived in a similar way and is given by [30]

if
*a*
and
*b*
are approaching, and
otherwise.
involves the energy
, which is identical to
(see Eq. (52)) except that
*W*
is replaced by
.

To determine the signal velocity Chow & Monaghan [30] (and Monaghan [122] in the non-relativistic case) start from the (local)
eigenvalues, and hence the wave velocities
and
*v*
of one-dimensional relativistic hydrodynamic flows. Again
considering particles
*a*
and
*b*
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
*a*
towards
*b*
and that from
*b*
to
*a*
. 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 [30]).

Chow & Monaghan [30] have demonstrated the performance of their Riemann problem
guided relativistic SPH algorithm by calculating several shock
tube problems involving ultra-relativistic speeds up to
*v*
= 0.9999. 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).

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 |