The hydrodynamic and MHD equations (either in Newtonian physics or in general relativity) constitute
nonlinear hyperbolic systems and, hence, smooth initial data can transform into discontinuous data (cross of
characteristics in the case of shocks) in a finite time during the evolution. As a consequence,
conventional finite-difference schemes (see, e.g., [218, 219
, 398
]) present important deficiencies when
dealing with such systems. Typically, first-order accurate schemes are much too dissipative across
discontinuities (excessive smearing) and second-order (or higher) schemes produce spurious
oscillations near discontinuities, which do not disappear under grid refinement. To avoid these effects,
standard finite-difference schemes have been conveniently modified in various ways to ensure
high-order, oscillation-free accurate representations of discontinuous solutions, as we discuss
next.
The idea of modifying the hydrodynamic equations by introducing artificial viscosity terms to damp the
amplitude of spurious oscillations near discontinuities was originally proposed by von Neumann and
Richtmyer [407] in the context of the (classical) Euler equations. The basic idea is to introduce a purely
artificial dissipative mechanism whose form and strength are such that the shock transition
becomes smooth, extending over a small number of intervals
of the space variable. In their
original work von Neumann and Richtmyer proposed the following expression for the viscosity
term:
with ,
being the fluid velocity,
the density,
the spatial interval, and
a
constant parameter whose value needs to be adjusted in every numerical experiment. This parameter
controls the number of zones in which shock waves are spread.
This type of generic recipe, with minor modifications, has been used in conjunction with standard
finite-difference schemes in all numerical simulations employing May and White’s formulation, mostly in the
context of gravitational collapse, as well as Wilson’s formulation. So, for example, in May and White’s
original code [247] the artificial viscosity term, obtained in analogy with the one proposed by von
Neumann and Richtmyer [407], is introduced in the equations accompanying the pressure, in the
form:
Further examples of similar expressions for the artificial viscosity terms, in the context of Wilson
formulation, can be found in, e.g., [411, 171
, 417]. In particular, a state-of-the-art formulation of the
artificial viscosity approach is reported in [19
]. Correspondingly, the interested reader is also directed to the
recent work by [86
, 20
, 108
] for details on diverse modern implementations of artificial viscosity terms in
nonconservative formulations of the GRMHD equations.
The main advantage of the artificial viscosity approach is its simplicity, which results in high
computational efficiency. Experience has shown, however, that this procedure is both problem dependent
and inaccurate for ultrarelativistic flows [291, 19
]. Furthermore, the artificial viscosity approach has the
inherent ambiguity of finding the appropriate form for
that introduces the necessary amount
of dissipation to reduce the spurious oscillations and, at the same time, avoids introducing
excessive smearing at discontinuities. In many instances both properties are difficult to achieve
simultaneously. A comprehensive numerical study of artificial-viscosity-induced errors in strong shock
calculations in Newtonian hydrodynamics (including also proposed improvements) was presented by
Noh [290].
In finite-difference schemes, convergence properties under grid refinement must be enforced to ensure that
the numerical results are correct (i.e., if a scheme with an order of accuracy is used, the global error of
the numerical solution has to tend to zero as
as the cell width
tends to zero). For
hyperbolic systems of conservation laws, schemes written in conservation form are preferred since, according
to the Lax–Wendroff theorem [216
], they guarantee that the convergence, if it exists, is to one of
the weak solutions of the original system of equations. Such weak solutions are generalized
solutions that satisfy the integral form of the conservation system. They are
classical
solutions (continuous and differentiable) away from discontinuities and have a finite number of
discontinuities.
For the sake of simplicity, let us consider in the following an initial value problem for a one-dimensional scalar hyperbolic conservation law,
and let us introduce a discrete numerical grid of spacetime points The class of all weak solutions is too wide in the sense that there is no uniqueness for the initial value
problem. The numerical method should, hence, guarantee convergence to the physically admissible solution.
This is the vanishing-viscosity–limit solution, that is, the solution when , of the “viscous version” of
the scalar conservation law, Equation (70
):
Mathematically, the solution one needs to search for is characterized by the entropy condition (in the language of fluids, the condition that the entropy of any fluid element should increase when crossing a discontinuity). The characterization of these entropy-satisfying solutions for scalar equations was given by Oleinik [301]. For hyperbolic systems of conservation laws it was developed by Lax [215].
The Lax–Wendroff theorem [216] cited above does not establish whether the method converges. To
guarantee convergence, some form of stability is required, as Lax first proposed for linear problems
(Lax equivalence theorem; see, e.g., [336]). Along this direction, the notion of total-variation
stability has proven very successful, although powerful results have only been obtained for
scalar conservation laws. The total variation of a solution at time , TV
, is defined
as
A numerical scheme is said to be TV-stable if TV is bounded for all
at any time for each
initial data. In the case of nonlinear, scalar conservation laws it can be proven that TV-stability is
a sufficient condition for convergence [218
], as long as the numerical schemes are written in
conservation form and have consistent numerical flux functions. Current research has focused
on the development of high-resolution numerical schemes in conservation form satisfying the
condition of TV-stability, such as the total variation diminishing (TVD) schemes [162
] (see
below).
Let us now consider the specific system of hydrodynamic equations as formulated in Equation (32) and
let us consider a single computational cell of our discrete spacetime. Let
be a region (simply connected)
of a given four-dimensional manifold
, bounded by a closed three-dimensional surface
. We
further take the three-surface
as the standard-oriented hyper-parallelepiped made up
of two spacelike surfaces
plus timelike surfaces
that join
the two temporal slices together. By integrating system (32
) over a domain
of a given
spacetime, the variation in time of the state vector
within
is given – keeping apart the
source terms – by the fluxes
through the boundary
. The integral form of system (32
)
is
where
Besides its convergence properties, a numerical scheme written in conservation form ensures that, in the absence of sources, the (physically) conserved quantities, according to the partial differential equations, are numerically conserved by the finite difference equations.
The computation of the time integrals of the interface fluxes appearing in Equation (76) is one of the
distinctive features of upwind HRSC schemes. One needs first to define the numerical fluxes, which are
recognized as approximations to the time-averaged fluxes across the cell interfaces, which depend on the
solution at those interfaces,
, during a timestep,
At the cell interfaces the flow can be discontinuous and, following the seminal idea of Godunov [155],
the numerical fluxes can be obtained by solving a collection of local Riemann problems, as is depicted in
Figure 2
. This is the approach followed by the Godunov-type methods [164
, 112
]. Figure 2
shows how the
continuous solution is locally averaged on the numerical grid, a process that leads to the appearance of
discontinuities at the cell interfaces. Physically, every discontinuity decays into three elementary waves of
any of the following type: shock waves, rarefaction waves, and contact discontinuities (see,
e.g., [398
]). The complete structure of the Riemann problem can be solved analytically (see [155] for
the solution in Newtonian hydrodynamics, [238
, 322
] in special-relativistic hydrodynamics,
and [340
, 150
] in special-relativistic MHD) and, accordingly, used to update the solution forward in
time.
For reasons of numerical efficiency and, particularly in multiple dimensions, the exact solution of the
Riemann problem is frequently avoided and linearized (approximate) Riemann solvers are preferred. These
solvers are based on the exact solution of Riemann problems corresponding to a linearized version of the
original system of equations. The spectral decomposition of the flux-vector Jacobian matrices is on
the basis of all solvers (extending ideas used for linear hyperbolic systems). After extensive
experimentation, the results achieved with approximate Riemann solvers are comparable to
those obtained with the exact solver (see [398] for a comprehensive overview of Godunov-type
methods, and [240
] for an excellent summary of approximate Riemann solvers in special-relativistic
hydrodynamics).
In the frame of the local characteristic approach, the numerical fluxes appearing in Equation (76) are
computed according to some generic flux formula that makes use of the characteristic information of the
system. For example, in Roe’s approximate Riemann solver [337
] it adopts the following functional
form:
For a purely linear system, Equation (80) provides the exact expression for the flux in terms of the
conserved variables. Therefore, the above expression for the Roe flux of conserved variables is the natural
extension (after linearization) of the upwind flux for characteristic variables (see, e.g., [218]), once the
quasilinear system is written in diagonal form, namely
The last term in the flux formula, Equation (80), represents the numerical viscosity of the scheme, and
it makes explicit use of the characteristic information of the Jacobian matrices of the system. This
information is used to provide the appropriate amount of numerical dissipation to obtain accurate
representations of discontinuous solutions without excessive smearing, avoiding, at the same time, the
growth of spurious numerical oscillations associated with the Gibbs phenomenon. In Equation (80
),
are, respectively, the eigenvalues and right eigenvectors of the Jacobian matrix of the flux
vector. Correspondingly, the quantities
are the jumps of the characteristic variables across
each characteristic field. They are obtained by projecting the jumps of the state-vector variables with the
left-eigenvector matrix:
The way the cell-reconstructed variables are computed determines the spatial order of accuracy of the
numerical algorithm and controls the amplitude of the local jumps at every cell interface. If these jumps are
monotonically reduced, the scheme provides more accurate initial guesses for the solution of the local
Riemann problems (the average values give only first-order accuracy). A wide variety of cell-reconstruction
procedures is available in the literature. Among the slope-limiter procedures (see, e.g., [398, 219
]) most
commonly used for TVD schemes [162] are the second-order, piecewise-linear reconstruction, introduced by
van Leer [403
] in the design of the MUSCL scheme (Monotonic Upstream Scheme for Conservation Laws),
and the third-order, piecewise-parabolic reconstruction developed by Colella and Woodward [80
] in their
Piecewise Parabolic Method (PPM). Since TVD schemes are only first-order accurate at local
extrema, alternative reconstruction procedures for which some growth of the total variation
is allowed have also been developed. Among those, we mention the total variation bounded
(TVB) schemes [377] and the essentially nonoscillatory (ENO) schemes [163] and extensions
thereof.
Alternatively, high-order methods for nonlinear hyperbolic systems have also been designed using flux
limiters rather than slope limiters, as in the Flux-Corrected Transport (FCT) scheme of Boris and
Book [60]. In this approach, the numerical flux consists of two pieces, a high-order flux (e.g., the
Lax–Wendroff flux) for smooth regions of the flow, and a low-order flux (e.g., the flux from some monotone
method) near discontinuities,
with the limiter
(see [398
, 219
] for
further details).
During the last few years most of the standard Riemann solvers developed in Newtonian fluid dynamics
have been extended to relativistic hydrodynamics: Eulderink [117], as discussed in Section 2.2.1, explicitly
derived a relativistic Roe’s Riemann solver [337]; Schneider et al. [345] carried out the extension of Harten,
Lax, van Leer, and Einfeldt’s (HLLE) method [164
, 112]; Martí and Müller [239
] extended the PPM
method of Woodward and Colella [420]; Wen et al. [410] extended Glimm’s exact Riemann solver; Dolezal
and Wong [100] put into practice Shu–Osher ENO techniques; Balsara [33] extended Colella’s
two-shock approximation; Donat et al. [101
] extended Marquina’s method [102
]. Recently,
much effort has been spent concerning the exact special relativistic Riemann solver and its
extension to multiple dimensions [238, 322, 333, 334
]. The interested reader is directed to [240
]
and references therein for a comprehensive description of such solvers in special-relativistic
hydrodynamics.
Upwind HRSC schemes are general enough to be applicable to any hyperbolic system as long as the
wave structure of the equations is known. As discussed before, the relativistic (and classical) MHD
system of equations show degeneracies in the eigenvectors of the flux-vector Jacobian matrices.
This fact makes it hazardous to solve them with linearized Riemann solvers based on the full
spectral decomposition of the flux-vector Jacobians. For the case of special-relativistic MHD,
approaches based on such full-wave decomposition have been put forward by [199, 34
, 198].
Advances in this front have been significant, as even the exact solution of the Riemann problem in
special-relativistic MHD has been obtained recently [340, 150]. In addition, there has been
a successful attempt by [24
] to develop a GRMHD code in which a full-wave decomposition
(Roe-type) Riemann solver based on a single, renormalized set of right and left eigenvectors has been
implemented. This set of eigenvectors is regular for any physical state, including degeneracies
(see [23
] for details). Such a Riemann solver is invoked in the code of [24
] after a (local) linear
coordinate transformation based on the procedure developed by [320
] that allows one to use
special-relativistic Riemann solvers in general relativity, and which has been properly extended to
include magnetic fields (see [24
] for details on such extension and Section 6.1 for details of the
procedure in the purely hydrodynamic case). A similar approach is followed in the GRMHD code of
Komissarov [200
].
Due to the numerical degeneracies of the GRMHD system most approaches to solve those equations use,
however, incomplete Riemann solvers, i.e., simpler alternative approaches to compute the numerical fluxes
such as the Harten–Lax–van Leer (HLL) single-state Riemann solver of [164] (see also [174] for an
improved HLL scheme at contact discontinuities (HLLC) for ideal special-relativistic MHD), or
high-order central (symmetric) schemes, which entirely sidestep the use of the Jacobian matrix
eigenvector structure. Such symmetric schemes, which have proven recently to yield results
with an accuracy comparable to those provided by full-wave decomposition Riemann solvers in
simulations involving both hydrodynamic and magnetohydrodynamic flows, are briefly discussed
next.
The use of high-order nonoscillatory symmetric (central) TVD schemes for solving hyperbolic systems of
conservation laws emerged in the mid 1980s [83, 338
, 427
, 287
] (see also [428] and [398
] and references
therein) as an alternative approach to upwind HRSC schemes. Symmetric schemes are based on either
high-order schemes (e.g., Lax–Wendroff) with additional dissipative terms [83
, 338, 427], or on
nonoscillatory high-order extensions of first-order central schemes (e.g., Lax–Friedrichs) [287, 226
]. One
of the nicest properties of central schemes is that they exploit the conservation form of the
Lax–Wendroff or Lax–Friedrichs schemes. Therefore, they yield the correct propagation speeds of all
nonlinear waves appearing in the solution. Furthermore, central schemes sidestep the use of
Riemann solvers, which results in enhanced computational efficiency, especially in multidimensional
problems. Its use is, thus, not limited to those systems of equations in which the characteristic
information is explicitly known or in which the Riemann problem is prohibitively expensive to
compute.
For illustrative purposes let us write the numerical flux function resulting in the fully-discrete central
scheme of Kurganov and Tadmor [211], to use in the conservation form algorithm given by
Equation (76
):
Conservative central schemes have been gradually developed during the last few years to reach a mature
status where a number of characteristic-information-free central schemes of high order can be applied to any
nonlinear hyperbolic system of conservation laws. The typical results obtained for the Euler equations show
a quality comparable to that of upwind HRSC schemes, at the expense of a small loss of sharpness of the
solution at discontinuities [398]. An up-to-date summary of the status and applications of this approach is
discussed in [398
, 211
, 392].
In recent years there have been various successful attempts to apply high-order central schemes to solve
the relativistic hydrodynamics equations as well, including those by [196] (Lax–Wendroff scheme with
conservative TVD dissipation terms), [89
] (Lax–Friedrichs or HLL schemes with third-order ENO
reconstruction algorithms), [19
] (nonoscillatory central differencing), and [230
, 361
] (semidiscrete central
scheme of Kurganov–Tadmor [211
]). On the other hand, symmetric schemes (such as HLL,
Kurganov–Tadmor, or Liu–Osher schemes) are being currently employed by a growing number of groups in
GRMHD [196
, 197
, 149
, 108
, 366
, 20
, 24
, 286
].
In the context of special and general-relativistic MHD, Koide et al. [196, 197
] applied a
second-order central scheme with nonlinear dissipation developed by [83
] to the study of black-hole
accretion and formation of relativistic jets, investigating issues such as the magnetic extraction of
rotational energy of the black hole [194
]. More recently [89
] assessed a state-of-the-art third-order,
convex, essentially nonoscillatory, central scheme [226] in multidimensional special-relativistic
hydrodynamics, later extended to relativistic MHD by [90
]. These authors obtained results as
accurate as those of upwind HRSC schemes in standard tests (shock tubes, shock reflection test).
Yet another central scheme has been considered by [19
, 20
] in one-dimensional special and
general-relativistic hydrodynamics and MHD, where results similar to those reported by [89
, 90] are
discussed. The scheme of Kurganov–Tadmor (see Equation (83
)) was assessed by [230
] in the
context of special-relativistic hydrodynamics, using standard numerical experiments such as
shock tubes, the shock reflection test, and the relativistic version of the flat-faced step test.
As for the other central schemes analyzed by [89
, 19
] the results were comparable to those
obtained by HRSC schemes based on Riemann solvers, even well inside the ultrarelativistic regime.
Lucas–Serrano et al. [230
] used high-order reconstruction procedures such as those provided by the PPM
scheme [80
] and the piecewise hyperbolic method (PHM) [102
], which proved essential for keeping the
inherent diffusion of central schemes at discontinuities at reasonably low levels. The scheme
also produced accurate results in the case of full general-relativistic hydrodynamic simulations
involving dynamic spacetimes, as shown by [361
] for oscillations of rapidly rotating neutron
stars and the merger of neutron-star binaries. The scheme of Kurganov–Tadmor is also the
adopted choice in the GRMHD simulations in dynamic spacetimes of [108
, 366
]. Finally, a
MUSCL-type scheme with HLL numerical fluxes is used in the code of [149
], which allows a systematic
investigation of GRMHD processes in accretion tori around black holes. Such HLL fluxes are
also the choice used in the recent approaches of [396
, 91
], which also implement a weighted,
essentially nonoscillatory (WENO) method to build up fifth-order convergent numerical codes. Such
high-order schemes may be suitable to solve the total (kinetic, thermal, and magnetic) energy
equation in GRMHD codes, which deal with the challenging regime posed by high-Mach number
flows.
It is worth emphasizing that early pioneer approaches in the field of relativistic hydrodynamics [291, 73]
used standard finite-difference schemes with artificial viscosity terms to stabilize the solution at
discontinuities. However, as discussed in Section 2.1.2, those approaches only succeeded in obtaining
accurate results for moderate values of the Lorentz factor (
). A key feature lacking in those
investigations was the use of a conservative approach for both the system of equations and, accordingly, for
the numerical schemes. In light of the results reported in recent investigations employing central
schemes [89, 19
, 230], it appears that this was the ultimate reason preventing the extension of early
computations to the ultrarelativistic regime.
The alternative of using high-order component-wise central schemes instead of upwind HRSC schemes
becomes apparent when the spectral decomposition of the hyperbolic system under study is unknown. The
straightforwardness of a central scheme makes its use very appealing, especially in multiple dimensions
where computational efficiency is an issue. Perhaps the most important example in relativistic astrophysics
is the system of general-relativistic MHD equations for which, despite some progress, has been achieved in
recent years (see, e.g., [34, 199, 24
, 23]), much more work is needed concerning their solution with
full-wave-decomposition HRSC schemes based upon Riemann solvers. Meanwhile, an obvious choice is the
use of central schemes.
Most “conservation laws” involve source terms on the right-hand side (RHS) of the equations. In
hydrodynamics, for instance, those terms arise when considering external forces such as gravity, which make
the RHS of the momentum and energy equations no longer zero (see Section 2). Other effects leading to the
appearance of source terms are radiative heat transfer (accounted for in the energy equation) and ionization
(resulting in a collection of nonhomogeneous continuity equations for the mass of each species, which is not
conserved separately). The incorporation of the source terms in the solution procedure is a common issue to
all numerical schemes considered so far. Since a detailed discussion on the numerical treatment of
source terms is beyond the scope of this article, we simply provide some basic information in
this section, directing the interested reader to [398, 219] and references therein for further
details.
There are, essentially, two ways of handling source terms. The first approach is based on unsplit methods
by which a single finite-difference formula advances the entire equation over one time step (as in
Equation (76)):
Correspondingly, the second approach is based on fractional-step or splitting methods. The basic idea is
to split the equation into different pieces (transport + sources) and apply appropriate methods
for each piece independently. In the first-order Godunov splitting, , the
operator
solves the homogeneous PDE in the first step to yield the intermediate value
. Then, in the second step, the operator
solves the ordinary differential equation
to yield the final value
. In order to achieve second-order accuracy (assuming
each independent method is second order) a common fractional-step method is the Strang splitting, where
. This method advances by a half timestep the solution for the ODE containing
the source terms, and by a full timestep the conservation law (the PDE) in between each source
step.
We note that in some cases the source terms may become stiff, as in phenomena that either occur on
much faster time scales than the hydrodynamic timestep , or act over much smaller spatial scales than
the grid resolution
. Stiff source terms may easily lead to numerical difficulties. The interested reader
is directed to [219] (and references therein) for further information on various approaches to overcome the
problems of stiff source terms.
http://www.livingreviews.org/lrr-2008-7 | ![]() This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 2.0 Germany License. Problems/comments to |