In the SPH method a finite set of extended Lagrangian particles replaces the continuum of hydrodynamical variables, the finite extent of the particles being determined by a smoothing function (the kernel) containing a characteristic length scale h . The main advantage of this 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 multi-dimensional applications, when, e.g., modelling regions containing large voids. Experience in Newtonian hydrodynamics shows that SPH produces very accurate results with a small number of particles ( or even less). However, if more than particles have to be used for certain problems, and self-gravity has to be included, the computational power, which grows as the square of the number of particles, may exceed the capabilities of current supercomputers. 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.
Reviews of the classical SPH equations are abundant in the literature (see, e.g., [187, 191] and references therein). The reader is addressed to  for a summary of comparisons between SPH and HRSC methods.
Recently, implementations of SPH to handle (special) relativistic (and even ultrarelativistic) flows have been developed (see, e.g.,  and references therein). However, SPH has been scarcely applied to simulate relativistic flows in curved spacetimes. Relevant references include [137, 146, 147, 271].
Following , let us describe the implementation of an SPH scheme in general relativity. Given a function , its mean smoothed value can be obtained from
where W is the smoothing kernel, h the smoothing length, and the volume element. The kernel must be differentiable at least once, and the derivative should be continuous to avoid large fluctuations in the force felt by a particle. Additional considerations for an appropriate election of the smoothing kernel can be found in . The kernel is required to satisfy a normalization condition,
which is assured by choosing , with , being a normalization function, and a standard spherical kernel.
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 being the collection of N particles where the functions are known. The previous representations then read:
with . These approximations can then be used to derive the SPH version of the general relativistic hydrodynamic equations. Explicit formulae are reported in . The time evolution of the final system of ODEs is performed in  using a second-order Runge-Kutta time integrator with adaptive time step. As in non-Riemann-solver-based finite volume schemes, in SPH simulations involving the presence of shock waves, artificial viscosity terms must be introduced as a viscous pressure term .
Recently, Siegler and Riffert  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  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., [160, 137, 146]). The conservative character of their scheme has allowed the modelling of ultrarelativistic flows including shocks with Lorentz factors as large as 1000.
Following  we illustrate the main ideas of spectral methods considering the quasi-linear one-dimensional scalar equation:
with u = u (t, x), and a constant parameter. In the linear case (), and assuming the function u to be periodic, spectral methods expand the function into a Fourier series:
From the numerical point of view, the series is truncated for a suitable value of k . Hence, Equation (59), 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 u (x, 0).
In the nonlinear case, , spectral methods proceed in a more convoluted way: First, the derivative of u 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 u in the configuration space, the scheme returns again to the Fourier space.
The particular set of trigonometric functions used for the expansion in Equation (60) 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 [110, 52] 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, as well as the associated violent phenomena of gravitational collapse and supernova explosion. They have developed a fully object-oriented library (based on the C++ computer language) called LORENE  to implement (multi-domain) spectral methods within spherical coordinates. A comprehensive summary of applications in general relativistic astrophysics is presented in . The most recent ones deal with the computation of quasi-equilibrium configurations of either synchronized or irrotational binary neutron stars in general relativity [112, 284, 283]. Such initial data are currently being used by fully relativistic, finite difference time-dependent codes (see Section 3.3.2) to simulate the coalescence of binary neutron stars.
The general relativistic hydrodynamic equations are expanded in a special form of the Taylor series:
with and denoting the first-order and second-order variation parameters. Using the above expressions, the time update then reads:
Combining the conservation form of the equations and the rearranged Taylor series expansion, allows us to rewrite the time update into standard matrix (residual) form, which can then be discretized using either standard finite difference or finite element methods .
The physical interpretation of the coefficients and is the foundation of the FDV method. The first-order parameter is proportional to , where is the convection Jacobian representing the change of convective motion. If the Lorentz factor remains constant in space and time, then . However, if the Lorentz factor between adjacent zones is large, . Similar assessments in terms of the Reynolds number can be provided for the diffusion and diffusion gradients, while the Froude number is used for the source term Jacobian . Correspondingly, the second-order FDV parameters are chosen to be exponentially proportional to the first-order ones.
Obviously, the main drawback of the FDV method is the dependence of the solution procedure on a large number of problem-dependent parameters, a drawback shared to some extent by artificial viscosity schemes. Richardson and Chung  have implemented the FDV method in a C++ code called GRAFSS (General Relativistic Astrophysical Flow and Shock Solver). The test problems they report are the special relativistic shock tube (problem 1 in the notation of ) and the Bondi accretion onto 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. Applications to non-ideal hydrodynamics and relativistic MHD are in progress .
|Numerical Hydrodynamics in General Relativity
José A. Font
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