Finite difference numerical schemes provide solutions of the
discretized version of the original system of partial
differential equations. Therefore, convergence properties under
grid refinement must be enforced on such schemes to ensure that
the numerical results are correct (i.e. the global error of the
numerical solution has to tend to zero as the cell width tends to
zero). For hyperbolic systems of conservation laws, schemes
written in
conservation form
are preferred as they guarantee that the convergence, if it
exists, is to one of the
weak solutions
of the original system of equations (Lax-Wendroff theorem
[116]). Such weak solutions are generalized solutions that satisfy
the integral form of the conservation system. They are classical
solutions (continuous and differentiable) in regions where they
are continuous and have a finite number of discontinuities.
Let us consider an initial value problem for a one-dimensional scalar hyperbolic conservation law,
Introducing a discrete numerical grid of space-time points
, an algorithm written in conservation form reads:
where
and
are the time-step and cell width respectively,
is a consistent numerical flux function (i.e.,
) and
is an approximation to 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 guarantee convergence to the
physically admissible solution
. This is the vanishing-viscosity limit solution, i.e., the
solution when
, of the ``viscous version'' of Eq. (48
):
Mathematically, this solution 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 the entropy-satisfying solutions for scalar equations was given by Oleinik [162]. For hyperbolic systems of conservation laws it was developed by Lax [115].
The Lax-Wendroff theorem [116] cited above does not establish whether the method converges. To
guarantee convergence, some form of stability is required, as Lax
proposed for linear problems (Lax equivalence theorem
; see, e.g., [182]). 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
, 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 non-linear
scalar conservation laws it can be proved that for numerical
schemes in conservation form with consistent numerical flux
functions, TV-stability is a sufficient condition for
convergence [117
]. Current research has focused on the development of
high-resolution numerical schemes in conservation form satisfying
the condition of TV-stability, e.g., the
total variation diminishing
(TVD) schemes. This issue is further discussed in Section
3.1.2
. Additionally, an important property that a numerical method
must satisfy is to be monotone. A scheme of the form
, with
and
two non-negative integers, is said to be monotone if all
coefficients
are positive or zero. It has been shown that, for scalar
conservation laws, monotone methods are TVD and satisfy a
discrete entropy condition. Therefore, they converge in a
non-oscillatory manner to the unique entropy solution. However,
monotone methods are at most first order accurate [117].
The hydrodynamic equations constitute a non-linear hyperbolic system and, hence, smooth initial data can give rise to discontinuous data (crossing of characteristics in the case of shocks) in a finite time during the evolution. As a consequence classical finite difference schemes present important deficiencies when dealing with such systems. Typically, first order accurate schemes are 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. Standard finite difference schemes have been conveniently modified in two ways to obtain high-order, oscillation-free accurate representations of discontinuous solutions as we discuss next.
Von Neumann and Richtmyer derived the following expression for the viscosity term:
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 recipe, with minor modifications, has been used
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
code [132] the artificial viscosity term, obtained in analogy with the one
originally proposed by von Neumann and Richtmyer [223], is introduced in the equations accompanying the pressure, in
the form
Further examples of equivalent expressions for the artificial
viscosity terms, in the context of Wilson formulation, can e.g.
be found in [226,
95
].
The main advantages of the artificial viscosity approach are: (i) It is straightforward to implement (compared to the HRSC schemes which need the characteristic fields of the equations), and (ii) it is computationally very efficient. Experience has shown, however, that this procedure is (i) problem dependent and (ii) inaccurate for ultrarelativistic flows [159]. Additionally, the artificial viscosity approach has the implicit difficulty 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 [158].
To focus the discussion let us consider the system as
formulated in Eq. (37). Let us consider a single computational cell of our discrete
spacetime. Let
be a region (simply connected) of the four-dimensional manifold
, bounded by a closed three-dimensional surface
. We 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 (37
) 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 (37
) is
which can be written in the following conservation form, well-adapted to numerical applications:
where
An important property of writing a numerical scheme in conservation form is 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 ``hat'' symbol appearing on the fluxes of Eq. (54) indicates the numerical fluxes. These 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 conditions
can be discontinuous and, following the seminal idea of
Godunov [84
], the numerical fluxes can be obtained by solving a collection
of local Riemann problems. This is depicted in Fig.
2
. The continuous solution is locally averaged on the numerical
grid, a process which 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 [84] for the solution in Newtonian hydrodynamics and [127] in special relativistic hydrodynamics) and, accordingly, used
to update the solution forward in time.
For reasons of efficiency and, particularly, in
multidimensions, 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 and, after extensive
experimentation, they are found to produce results comparable to
those obtained with the exact solver (see [126] for a summary of such approximate solvers in special
relativistic hydrodynamics).
In the frame of the local characteristic approach the numerical fluxes are computed according to some generic flux-formula which makes use of the characteristic information of the system. For example, in Roe's approximate Riemann solver it adopts the following functional form:
where
and
represent the values of the primitive variables at the left and
right sides, respectively, of the corresponding 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 the numerical algorithm
and controls the local jumps at every interface. If these jumps
are monotonically reduced, the scheme provides more accurate
initial guesses for the solution of the local Riemann problems. A
wide variety of cell reconstruction procedures is available in
the literature. Among the most popular slope limiter procedures
for TVD schemes [88] are the second order piecewise linear reconstruction,
introduced by van Leer [219
] 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 [52
] in their Piecewise Parabolic Method (PPM). High order piecewise
polynomial functions are also available for Essentially
Non-Oscillatory (ENO) schemes [89].
The last term in the flux-formula 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.
Hence,
are, respectively, the eigenvalues and right-eigenvectors of the
Jacobian matrix, and 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-eigenvectors
matrix:
The ``tilde'' 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
classical
Riemann solvers developed in fluid dynamics have been extended
to relativistic hydrodynamics: Eulderink [64], as discussed in Section
2.2.1, has explicitly derived a relativistic Roe Riemann solver [183
], Schneider et al. [190] carried out the extension of Einfeldt's HLLE method [90,
61], Martí and Müller [128
] extended the PPM method of Woodward and Colella [233], Wen et al. [224] extended Glimm's method, Dolezal and Wong [55] put into practice Shu-Osher ENO techniques, Balsara [16] extended Colella's two-shock approximation and, Donat et
al. [56
] extended Marquina's method [57
]. The interested reader is referred to [126] for a comprehensive description of such solvers in special
relativistic hydrodynamics.
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Numerical Hydrodynamics in General Relativity
José A. Font http://www.livingreviews.org/lrr-2000-2 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |