The hydrodynamic equations (either in Newtonian physics or in
general relativity) constitute a nonlinear hyperbolic system and,
hence, smooth initial data can transform into discontinuous data
(the crossing of characteristics in the case of shocks) in a
finite time during the evolution. As a consequence, classical
finite difference schemes (see, e.g., [152,
287
]) 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 as the grid is refined.
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.
with
,
v
being the fluid velocity,
the density,
the spatial interval, and
k
a constant parameter whose value is conveniently 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 conjuction 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 [172] the artificial viscosity term, obtained in analogy with the one
proposed by von Neumann and Richtmyer [295], 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., [300,
123
]. A state-of-the-art formulation of the artificial viscosity
approach is reported in [13
].
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 [208,
13
]. Furthermore, the artificial viscosity approach has the
inherent ambiguity of finding the appropriate form for
Q
that introduces the necessary amount of dissipation to reduce
the spurious oscillations and, at the same time, avoids
introducing excessive smearing in the 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 [207].
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 space-time
points
. An explicit algorithm written in conservation form updates the
solution from time
to the next time level
as
where
is a consistent numerical flux function (i.e.,
) of
p
+
q
+1 arguments and
and
are the time step and cell width respectively. Furthermore,
is an approximation of the average of
u
(x,
t) within the numerical cell
:
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 (39
):
Mathematically, the solution one needs to search for is characterized by the so-called entropy condition (in the language of fluids, the condition that the entropy of any fluid element should increase when running into a discontinuity). The characterization of these entropy-satisfying solutions for scalar equations was given by Oleinik [212]. For hyperbolic systems of conservation laws it was developed by Lax [149].
The Lax-Wendroff theorem [150] 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., [241]). 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 proved that TV-stability is a
sufficient condition for convergence [152], 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 so-called
total variation diminishing
(TVD) schemes [115
] (see below).
Let us now consider the specific system of hydrodynamic
equations as formulated in Equation (30), 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 3-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 (30
) 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 (30
) is
which can be written in the following conservation form, well-adapted to numerical applications:
where
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 (45) is one of the distinctive features of upwind HRSC schemes. One
needs first to define the so-called 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 time step,
At the cell interfaces the flow can be discontinuous and,
following the seminal idea of Godunov [108], 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 so-called Godunov-type
methods [117
,
75
]. 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: a shock wave, a
rarefaction wave, and a contact discontinuity. The complete
structure of the Riemann problem can be solved analytically
(see [108] for the solution in Newtonian hydrodynamics and [165
,
231
] in special relativistic hydrodynamics) and, accordingly, used
to update the solution forward in time.
For reasons of numerical efficiency and, particularly in
multi-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. After extensive experimentation,
the results achieved with approximate Riemann solvers are
comparable to those obtained with the exact solver (see [287] for a comprehensive overview of Godunov-type methods,
and [164
] 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 (45) 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 [242
] it adopts the following functional form:
where
and
are the values of the primitive variables at the left and right
sides, respectively, of a given cell interface. They are obtained
from the cell centered quantities after a suitable monotone
reconstruction procedure.
The way these 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., [287,
153
]) most commonly used for TVD schemes [115] are the second order, piecewise-linear reconstruction,
introduced by van Leer [290
] 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 [58
] 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 [268] and the
essentially non-oscillatory
(ENO) schemes [116].
Alternatively, high-order methods for nonlinear hyperbolic
systems have also been designed using flux limiters rather than
slope limiters, as in the FCT scheme of Boris and Book [46]. 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 [287
,
153
] for further details).
The last term in the flux-formula, Equation (49), 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 (49
),
are the eigenvalues and right-eigenvectors of the Jacobian
matrix of the flux vector, respectively. Correspondingly, the
quantities
are the jumps of the so-called characteristic variables across
each characteristic field. They are obtained by projecting the
jumps of the state vector variables with the left-eigenvectors
matrix:
The ``tilde'' in Equations (49) and (50
) indicates that the corresponding fields are averaged at the
cell interfaces from the left and right (reconstructed)
values.
During the last few years most of the standard Riemann solvers
developed in Newtonian fluid dynamics have been extended to
relativistic hydrodynamics: Eulderink [78], as discussed in Section
2.2.1, explicitly derived a relativistic Roe Riemann solver [242
]; Schneider et al. [250] carried out the extension of Harten, Lax, van Leer, and
Einfeldt's (HLLE) method [117,
75]; Martí and Müller [166
] extended the PPM method of Woodward and Colella [306]; Wen et al. [297] extended Glimm's exact Riemann solver; Dolezal and Wong [68] put into practice Shu-Osher ENO techniques; Balsara [19] extended Colella's two-shock approximation, and Donat et
al. [69
] extended Marquina's method [70
]. Recently, much effort has been spent concerning the exact
special relativistic Riemann solver and its extension to
multi-dimensions [165,
231,
238,
239
]. The interested reader is addressed to [164] and references therein for a comprehensive description of such
solvers in special relativistic hydrodynamics.
In the context of special and general relativistic MHD, Koide
et al. [141,
142
] applied a second-order central scheme with nonlinear
dissipation developed by Davis [61
] to the study of black hole accretion and formation of
relativistic jets. One-dimensional test simulations in special
relativistic hydrodynamics performed by the author and
coworkers [92
] using the SLIC (slope limiter centred) scheme (see [287
] for details) showed its capabilities to yield satisfactory
results, comparable to those obtained by HRSC schemes based on
Riemann solvers, even well inside the ultrarelativistic regime.
The slopes of the original central scheme were limited using
high-order reconstruction procedures such as PPM [58
], which was essential to keep the inherent diffusion of central
schemes at discontinuities at reasonable levels. Very recently,
Del Zanna and Bucciantini [63
] assessed a third-order convex essentially non-oscillatory
central scheme in multi-dimensional special relativistic
hydrodynamics. Again, 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
assessed by [13
] in one-dimensional special and general relativistic
hydrodynamics, where similar results to those of [63
] are reported. These authors also validate their central scheme
in simulations of spherical accretion onto a Schwarzschild black
hole, and further provide a comparison with an artificial
viscosity based scheme.
It is worth emphasizing that early pioneer approaches in the
field of relativistic hydrodynamics [208,
54] 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 the
numerical schemes. A posteriori, and in the light of the results
reported by [63,
13,
92], it appears that this was the ultimate reason preventing the
extension of the early computations to the ultrarelativistic
regime.
The alternative of using high-order central schemes instead of
upwind HRSC schemes becomes apparent when the spectral
decomposition of the hyperbolic system under study is not known.
The straightforwardness of a central scheme makes its use very
appealing, especially in multi-dimensions where computational
efficiency is an issue. Perhaps the most important example in
relativistic astrophysics is the system of (general) relativistic
MHD equations. Despite some progress in recent years (see, e.g.,
[20,
143]), much more work is needed concerning their solution with HRSC
schemes based upon Riemann solvers. Meanwhile, an obvious choice
is the use of central schemes [141,
142
].
There are, essentially, two basic 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 (45)):
The temporal order of this basic algorithm can be improved by introducing successive sub-steps to perform the time update (e.g., predictor-corrector, Shu and Osher's conservative high order Runge-Kutta schemes, etc.)
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
. Therefore, this method advances by a half time step the
solution for the ODE containing the source terms, and by a full
time step 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 timescales
than the hydrodynamic time step
, 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 [153] (and references therein) for further information on various
approaches to overcome the problems of stiff source terms.
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Numerical Hydrodynamics in General Relativity
José A. Font http://www.livingreviews.org/lrr-2003-4 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |