4.3 Relativistic beam scheme4 Other Developments4.1 Van Putten's approach

4.2 Relativistic SPH 

Besides finite volume schemes, another completely different method is widely used in astrophysics for integrating the hydrodynamic equations. This method is Smoothed Particle Hydrodynamics, or SPH for short [100, 63, 121Jump To The Next Citation Point In The Article]. The fundamental idea of SPH is to represent a fluid by a Monte Carlo sampling of its mass elements. The motion and thermodynamics of these mass elements is then followed as they move under the influence of the hydrodynamics equations. Because of its Lagrangian nature there is no need within SPH for explicit integration of the continuity equation, but in some implementations of SPH this is done nevertheless for certain reasons. As both the equation of motion of the fluid and the energy equation involve continuous properties of the fluid and their derivatives, it is necessary to estimate these quantities from the positions, velocities and internal energies of the fluid elements, which can be thought of as particles moving with the flow. This is done by treating the particle positions as a finite set of interpolating points where the continuous fluid variables and their gradients are estimated by an appropriately weighted average over neighboring particles. Hence, SPH is a free-Lagrange method, i.e., spatial gradients are evaluated without the use of a computational grid.

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

The SPH equations for special relativistic flows have been first formulated by Monaghan [120Jump To The Next Citation Point In The Article]. For such flows the SPH equations given in Section  9.5 can be taken over except that each SPH particle a carries tex2html_wrap_inline6073 baryons instead of mass tex2html_wrap_inline6075  [120Jump To The Next Citation Point In The Article, 30Jump To The Next Citation Point In The Article]. Hence, the rest mass of particle a is given by tex2html_wrap_inline6079, where tex2html_wrap_inline6081 is the baryon rest mass (if the fluid is made of baryons). Transforming the notation used in [30Jump To The Next Citation Point In The Article] to ours, the continuity equation, the momentum and the total energy equations for particle a are given by (unit of velocity is c)

  equation804

  equation811

and

  equation820

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

equation829

is the baryon number density,

  equation832

the momentum per particle, and

  equation838

the total energy per particle (all measured in the laboratory frame). The momentum density tex2html_wrap_inline6091, the energy density tex2html_wrap_inline5711 (measured in units of the rest mass energy density), and the specific enthalpy h are defined in Section  2.1 . tex2html_wrap_inline6097 and tex2html_wrap_inline6099 are the SPH dissipation terms, and tex2html_wrap_inline6101 denotes the gradient of the kernel tex2html_wrap_inline6103 (see Section  9.5 for more details).

Special relativistic flow problems have been simulated with SPH by [90Jump To The Next Citation Point In The Article, 80Jump To The Next Citation Point In The Article, 102Jump To The Next Citation Point In The Article, 104, 30Jump To The Next Citation Point In The Article, 164Jump To The Next Citation Point In The Article]. Extensions of SPH capable of treating general relativistic flows have been considered by [80, 89Jump To The Next Citation Point In The Article, 164Jump To The Next Citation Point In The Article]. 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 [102Jump To The Next Citation Point In The Article] and Laguna et al. [89Jump To The Next Citation Point In The Article] 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 [102Jump To The Next Citation Point In The Article] 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 [164Jump To The Next Citation Point In The Article] use a consistent artificial viscosity pressure added to the fluid pressure. Siegler & Riffert [164Jump To The Next Citation Point In The Article] have also formulated the hydrodynamic equations in conservation form.

Monaghan [122Jump To The Next Citation Point In The Article] 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 [30Jump To The Next Citation Point In The Article] 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 [155Jump To The Next Citation Point In The Article], as well as in its relativistic variant proposed by Eulerdink [49Jump To The Next Citation Point In The Article, 50Jump To The Next Citation Point In The Article] (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. (26Popup Equation) and (27Popup Equation)). Monaghan [122Jump To The Next Citation Point In The Article] realized that an appropriate form of the dissipative terms tex2html_wrap_inline6097 and tex2html_wrap_inline6099 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 tex2html_wrap_inline6113 (see below), and that equivalent to the jump across characteristics is a jump in the relevant physical variable. For the artificial viscosity tensor, tex2html_wrap_inline6097, Monaghan [122Jump To The Next Citation Point In The Article] 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 [30Jump To The Next Citation Point In The Article] proposed for tex2html_wrap_inline6097 in the relativistic case the form

  equation883

when particles a and b are approaching, and tex2html_wrap_inline6127 otherwise. Here K = 0.5 is a dimensionless parameter, which is chosen to have the same value as in the non-relativistic case [122Jump To The Next Citation Point In The Article]. tex2html_wrap_inline6131 is the average baryon number density, which has to be present in (53Popup Equation), because the pressure terms in the summation of (90Popup Equation) have an extra density in the denominator arising from the SPH interpolation. Furthermore,

  equation903

is the unit vector from b to a, and

  equation911

where

  equation916

Using instead of tex2html_wrap_inline6137 (see Eq. (51Popup Equation)) the modified momentum tex2html_wrap_inline6139, which involves the line of sight velocity tex2html_wrap_inline6141, guarantees that the viscous dissipation is positive definite [30Jump To The Next Citation Point In The Article].

The dissipation term in the energy equation is derived in a similar way and is given by [30Jump To The Next Citation Point In The Article]

  equation929

if a and b are approaching, and tex2html_wrap_inline6147 otherwise. tex2html_wrap_inline6099 involves the energy tex2html_wrap_inline6151, which is identical to tex2html_wrap_inline6153 (see Eq. (52Popup Equation)) except that W is replaced by tex2html_wrap_inline6157 .

To determine the signal velocity Chow & Monaghan [30Jump To The Next Citation Point In The Article] (and Monaghan [122Jump To The Next Citation Point In The Article] in the non-relativistic case) start from the (local) eigenvalues, and hence the wave velocities tex2html_wrap_inline6159 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 [30Jump To The Next Citation Point In The Article]).

Chow & Monaghan [30Jump To The Next Citation Point In The Article] 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).



4.3 Relativistic beam scheme4 Other Developments4.1 Van Putten's approach

image Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller
http://www.livingreviews.org/lrr-1999-3
© Max-Planck-Gesellschaft. ISSN 1433-8351
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