The Lagrangian particle method SPH, derived independently by Lucy [231] and Gingold and Monaghan [152], has shown successful performance in modelling fluid flows in astrophysics and cosmology. Most studies to date consider Newtonian flows and gravity, enhanced with the inclusion of the fluid self-gravity.
In the SPH method a finite set of extended Lagrangian particles replaces the continuum of
hydrodynamic variables, the finite extent of the particles being determined by a smoothing function
(the kernel) containing a characteristic length scale . SPH-discretized representations of
the equations of motion can be obtained after the introduction of mean-smoothed values for
functions and for the gradient and divergence operators, as is briefly outlined below. Reviews of the
classical SPH equations are abundant in the literature (see, e.g., [267, 271
] and references
therein).
The main advantage of the SPH method is that it does not require a computational grid, avoiding mesh
tangling and distortion. Hence, compared to grid-based finite-volume methods, SPH avoids wasting
computational power in multidimensional applications, when, e.g., modelling regions containing large voids.
Among the limitations of SPH we note the difficulties in modelling systems with extremely different
characteristic lengths and the fact that boundary conditions usually require a more involved
treatment than in finite-volume schemes. Comparisons between SPH and HRSC methods have been
reported by various authors (see, e.g., [271, 4
] and references therein). In particular, the recent
comparison study of state-of-the-art hydrodynamic codes carried out by Agertz et al. [4] has revealed
fundamental differences between SPH and grid methods in modelling interacting multiphase fluids.
More precisely, dynamic instabilities such as Kelvin–Helmholtz or Rayleigh–Taylor are much
better resolved by Eulerian grid-based methods than by SPH techniques, the reason being the
appearance of spurious pressure forces on particles in regions where shocks or steep gradients
appear.
In the past few years, implementations of SPH to handle (special) relativistic (and even ultrarelativistic)
flows have been developed (see, e.g., [79] and references therein). However, SPH has been so
far scarcely applied in simulating relativistic flows in curved spacetimes. Relevant references
include [189
, 212
, 213, 381
, 300
, 122
, 299
].
Following [212], let us describe the implementation of an SPH scheme in general relativity.
Given a function
, its mean smoothed value
can be obtained
from
The smooth approximation of gradients of scalar functions can be written as
and the approximation of the divergence of a vector reads Discrete representations of these procedures are obtained after introducing the number density
distribution of particles , with
the collection of
-particles
where the functions are known. The previous representations then read:
Recently, Siegler and Riffert [381] have developed a Lagrangian conservation form of the
general-relativistic hydrodynamic equations for perfect fluids (with artificial viscosity) in arbitrary
background spacetimes. Within that formulation these authors [381
] have built a general-relativistic SPH
code using the standard SPH formalism as known from Newtonian fluid dynamics (in contrast to
previous approaches, e.g., [232, 189, 212
]). The conservative character of their scheme has
allowed the modelling of ultrarelativistic flows including shocks with Lorentz factors as large as
1000.
The basic principle underlying spectral methods consists of transforming the partial differential
equations into a system of ordinary differential equations by means of expanding the solution in a
series on a complete basis. The mathematical theory of these schemes is presented in [156, 71
]
(see also [159
] for a recent Living Review article on spectral methods for numerical relativity).
Spectral methods are particularly well suited to the solution of elliptic and parabolic equations.
Good results can also be obtained for hyperbolic equations as long as no discontinuities appear
in the solution. When a discontinuity is present, some amount of artificial viscosity must be
added to avoid spurious oscillations. In the specific case of relativistic problems, where coupled
systems of elliptic equations (i.e., the Einstein constraint equations) and hyperbolic equations
(i.e., hydrodynamics) must be solved, an interesting strategy is to use spectral methods to
solve the elliptic equations and HRSC schemes for the hyperbolic ones. Using such combined
methods, first results were obtained in one-dimensional supernova collapse simulations, both in the
framework of a tensor-scalar theory of gravitation [292, 294
] and in general relativity [293].
Multidimensional approaches for core collapse and studies of neutron star dynamics are available
in [97
, 99
].
Following [55] we illustrate the main ideas of spectral methods considering the quasi-linear
one-dimensional scalar equation:
From the numerical point of view, the series is truncated for a suitable value of . Hence,
Equation (92
), with
, can be rewritten as
Finding a solution of the original equation is then equivalent to solving an “infinite” system of ordinary
differential equations, where the initial values of the coefficients and
are given by the Fourier
expansion of
.
In the nonlinear case, , spectral methods proceed in a more convoluted way: first, the derivative
of
is computed in the Fourier space. Then, a step back to the configuration space is taken through an
inverse Fourier transform. Finally, after multiplying
by
in the configuration space, the scheme
returns again to the Fourier space.
The particular set of trigonometric functions used for the expansion in Equation (93) is chosen because
it automatically fulfills the boundary conditions, and because a fast transform algorithm is available (the
latter is no longer an issue for today’s computers). If the initial or boundary conditions are not
periodic, Fourier expansion is no longer useful because of the presence of a Gibbs phenomenon at
the boundaries of the interval. Legendre or Chebyshev polynomials are, in this case, the most
common base of functions used in the expansions (see [156, 71] for a discussion on the different
expansions).
Extensive numerical applications using (pseudo) spectral methods have been undertaken by the LUTH
Relativity Group at the Observatoire de Paris in Meudon. The group has focused on the study of
compact objects, providing stationary initial data for single and binary compact stars and black
holes, as well as performing dynamic studies of gravitational collapse to neutron stars and black
holes. The Meudon group has developed a fully object-oriented library (based on the C++
computer language) called LORENE [227] where (multiple domain) spectral methods have been
implemented within spherical coordinates. This library is currently being used by a growing number of
numerical relativity groups worldwide, as can be inferred from the comprehensive summary of
applications in general-relativistic astrophysics presented in [159] (see also [55] for an earlier review).
Additional numerical relativity codes based on spectral methods have been developed by Ansorg and
collaborators, who have built remarkably accurate stationary data for relativistic stars and black
holes [21, 22], by the Cornell/Caltech group, who are focused on the black-hole–binary problem, and
by Faber et al. [122
], who have investigated the black-hole–neutron-star binary system for a
conformally-flat metric employing LORENE and an SPH code to solve for the hydrodynamics (see
below).
The finite-element method is the preferred choice for solving multidimensional structural-engineering
problems since the late 1960s [436]. First steps in the development of the finite-element method for
modeling astrophysical flows in general relativity are discussed by Meier [253]. The method, designed in a
fully covariant manner, is valid not only for the hydrodynamic equations but also for the Einstein and
electromagnetic field equations. The most unique aspect of the approach presented in [253
] is
that the grid can be arbitrarily extended into the time dimension. Therefore, the standard
finite-element method generalizes to a four-dimensional covariant technique on a spacetime
grid, with the engineer’s “isoparametric transformation” becoming the generalized Lorentz
transform. At present, the method has shown its suitability for accurately computing the equilibrium
stellar structure of Newtonian polytropes, either spherical or rotating. The main limitation
of the finite-element method, as Meier points out [253], is that it is generally fully implicit,
which results in severe computer time and memory limitations for three and four-dimensional
problems.
Richardson and Chung [335] proposed recently the flow field-dependent variation (FDV) method as a
new approach for general relativistic (non ideal) hydrodynamics computations. In the FDV method,
parameters characteristic of the flow field are computed in order to guide the numerical scheme toward a
solution. The basic idea is to expand the conservation flow variables into a Taylor series in terms
of the FDV parameters, which are related to variations of physical parameters such as the
Lorentz factor, the relativistic Reynolds number and the relativistic Froude number. The main
drawback of the FDV method is the dependence of the solution procedure on a large number of
problem-dependent parameters. Richardson and Chung [335] implemented the FDV method
in a C++ code called GRAFSS (General-Relativistic Astrophysical Flow and Shock Solver).
The test problems reported are the special relativistic shock tube (problem 1 in the notation
of [239
]) and the Bondi accretion on to a Schwarzschild black hole. While their method yields the
correct wave propagation, the numerical solution near discontinuities has considerably more
diffusion than with upwind HRSC schemes. Nevertheless, the generality of the approach is worth
considering.
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