The von Neumann stability analysis of the interior algorithm
linearizes the equations, while assuming a uniform infinite grid,
and checks that the discrete Fourier modes do not grow
exponentially. There is an additional stability condition that a
boundary introduces into this analysis. Consider the
one-dimensional case. Normally the mode
, with
k
real, is not included in the von Neumann analysis. However, if
there is a boundary to the right on the
x
-axis, one can legitimately prescribe such a mode (with
k
>0) as initial data, so that its stability must be checked. In
the case of an additional boundary to the left, the
Ryaben'kii-Godunov theory allows separate investigation of the
interior stability and the stability of each individual
boundary [128].
The correct physical formulation of any asymptotically flat, radiative Cauchy problem also requires boundary conditions at infinity. These conditions must ensure not only that the total energy and the energy loss by radiation are both finite, but must also ensure the proper 1/ r asymptotic falloff of the radiation fields. However, when treating radiative systems computationally, an outer boundary must be established artificially at some large but finite distance in the wave zone, i.e. many wavelengths from the source. Imposing an accurate radiation boundary condition at a finite distance is a difficult task even in the case of a simple radiative system evolving on a fixed geometric background. The problem is exacerbated when dealing with Einstein's equation.
Nowhere is the boundary problem more acute than in the computation of gravitational radiation produced by black holes. The numerical study of a black hole spacetime by means of a pure Cauchy evolution involves inner as well as outer grid boundaries. The inner boundary is necessary to avoid the topological complications and singularities introduced by a black hole. For multiple black holes, the inner boundary consists of disjoint pieces. W. Unruh (see [145]) initially suggested the commonly accepted strategy for Cauchy evolution of black holes. An inner boundary located at (or near) an apparent horizon is used to excise the singular interior region. Later (see Sec. 4.8) I discuss a variation of this strategy based upon matching to a characteristic evolution in the inner region.
First, consider the outer boundary problem, in which
Cauchy-characteristic matching has a natural application. In the
Cauchy treatment of such a system, the outer grid boundary is
located at some finite distance, normally many wavelengths from
the source. Attempts to use compactified Cauchy hypersurfaces
which extend to spatial infinity have failed because the phase of
short wavelength radiation varies rapidly in spatial
directions [93]. Characteristic evolution avoids this problem by approaching
infinity along phase fronts.
When the system is nonlinear and not amenable to an exact
solution, a finite outer boundary condition must necessarily
introduce spurious physical effects into a Cauchy evolution. The
domain of dependence of the initial Cauchy data in the region
spanned by the computational grid would shrink in time along
ingoing characteristics unless data on a worldtube traced out by
the outer grid boundary is included as part of the problem. In
order to maintain a causally sensible evolution, this worldtube
data must correctly substitute for the missing Cauchy data which
would have been supplied if the Cauchy hypersurface had extended
to infinity. In a scattering problem, this missing exterior
Cauchy data might, for instance, correspond to an incoming pulse
initially outside the outer boundary. In a problem where the
initial radiation fields are confined to a compact region inside
the boundary, this missing Cauchy data are easy to state when
dealing with a constraint free field, such as a scalar field
where the Cauchy data outside the boundary would be
. However, the determination of Cauchy data for general
relativity is a global elliptic constraint problem so that there
is no well defined scheme to confine it to a compact region.
Furthermore, even if the data were known on a complete initial
hypersurface extending to infinity, it would be a formidable
nonlinear problem to correctly pose the associated data on the
outer boundary. The only formulation of Einstein's equations
whose Cauchy initial-boundary value problem has been shown to
have a unique solution is a symmetric hyperbolic formulation due
to Friedrich and Nagy [61]. However, their mathematical treatment does not suggest how to
construct a consistent evolution algorithm.
It is common practice in computational physics to impose some
artificial boundary condition (ABC), such as an outgoing
radiation condition, in an attempt to approximate the proper data
for the exterior region. This ABC may cause partial reflection of
an outgoing wave back into the system [101,
93,
87
,
118
], which contaminates the accuracy of the interior evolution and
the calculation of the radiated waveform. Furthermore, nonlinear
waves intrinsically backscatter, which makes it incorrect to try
to entirely eliminate incoming radiation from the outer region.
The errors introduced by these problems are of an analytic
origin, essentially independent of computational discretization.
In general, a systematic reduction of this error can only be
achieved by simultaneously refining the discretization and moving
the computational boundary to larger and larger radii. This is
computationally very expensive, especially for three-dimensional
simulations.
A traditional outer boundary condition for the wave equation
is the Sommerfeld condition. For a 3D scalar field this takes the
form
, where
. This condition is
exact
for a linear wave with spherically symmetric data and boundary.
In that case, the exact solution is
and the Sommerfeld condition eliminates the incoming wave
.
Much work has been done on formulating boundary conditions,
both exact and approximate, for linear problems in situations
that are not spherically symmetric and in which the Sommerfeld
condition would be inaccurate. These boundary conditions are
given various names in the literature, e.g. absorbing or
non-reflecting. A variety of successful implementations of ABC's
have been reported for linear problems. See the recent
articles [66,
118,
148
,
120
,
27
] for a general discussion of ABC's.
Local ABC's have been extensively applied to linear problems
with varying success [101,
56
,
22
,
147
,
87
,
33,
94]. Some of these conditions are local approximations to exact
integral representations of the solution in the exterior of the
computational domain [56
], while others are based on approximating the dispersion
relation of the so-called one-way wave equations [101,
147]. Higdon [87] showed that this last approach is essentially equivalent to
specifying a finite number of angles of incidence for which the
ABC's yield perfect transmission. Local ABC's have also been
derived for the linear wave equation by considering the
asymptotic behavior of outgoing solutions [22], which generalizes the Sommerfeld outgoing radiation condition.
Although such ABC's are relatively simple to implement and have a
low computational cost, their final accuracy is often limited
because the assumptions made about the behavior of the waves are
rarely met in practice [66
,
148
].
The disadvantages of local ABC's have led some workers to
consider exact nonlocal boundary conditions based on integral
representations of the infinite domain problem [146,
66,
148
]. Even for problems where the Green's function is known and
easily computed, such approaches were initially dismissed as
impractical [56
]; however, the rapid increase in computer power has made it
possible to implement exact nonlocal ABC's for the linear wave
equation and Maxwell's equations in 3D [48
,
78]. If properly implemented, this kind of method can yield
numerical solutions which converge to the exact infinite domain
problem in the continuum limit, keeping the artificial boundary
at a fixed distance. However, due to nonlocality, the
computational cost per time step usually grows at a higher power
with grid size (
per time step in three dimensions) than in a local
approach [66
,
48,
148
].
The extension of ABC's to
nonlinear
problems is much more difficult. The problem is normally treated
by linearizing the region between the outer boundary and
infinity, using either local or nonlocal linear ABC's [148,
120]. The neglect of the nonlinear terms in this region introduces
an unavoidable error at the analytic level. But even larger
errors are typically introduced in prescribing the outer boundary
data. This is a subtle global problem because the correct
boundary data must correspond to the continuity of fields and
their normal derivatives when extended across the boundary into
the linearized exterior. This is a clear requirement for any
consistent boundary algorithm, since discontinuities in the field
or its derivatives would otherwise act as spurious sheet source
on the boundary, thereby contaminating both the interior and the
exterior evolutions. But the fields and their normal derivatives
constitute an overdetermined set of data for the linearized
exterior problem. So it is necessary to solve a global linearized
problem, not just an exterior one, in order to find the proper
data. The designation ``exact ABC'' is given to an ABC for a
nonlinear system whose only error is due to linearization of the
exterior. An exact ABC requires the use of global techniques,
such as the difference potential method, to eliminate back
reflection at the boundary [148].
To date there have been only a few applications of ABC's to
strongly nonlinear problems [66]. Thompson [144
] generalized a previous nonlinear ABC of Hedstrom [86] to treat 1D and 2D problems in gas dynamics. These boundary
conditions performed poorly in some situations because of their
difficulty in adequately modeling the field outside the
computational domain [144,
66]. Hagstrom and Hariharan [82] have overcome these difficulties in 1D gas dynamics by a clever
use of Riemann invariants. They proposed a heuristic
generalization of their local ABC to 3D, but this has not yet
been implemented.
In order to reduce the level of approximation at the analytic level, an artificial boundary for an nonlinear problem must be placed sufficiently far from the strong-field region. This sharply increases the computational cost in multidimensional simulations [56]. There seems to be no numerical method which converges (as the discretization is refined) to the infinite domain exact solution of a strongly nonlinear wave problem in multidimensions, while keeping the artificial boundary fixed.
Cauchy-characteristic matching is a strategy that eliminates this nonlinear source of error. In CCM, Cauchy and characteristic evolution algorithms are pasted together in the neighborhood of a worldtube to form a global evolution algorithm. The characteristic algorithm provides an outer boundary condition for the interior Cauchy evolution, while the Cauchy algorithm supplies an inner boundary condition for the characteristic evolution. The matching worldtube provides the geometric framework necessary to relate the two evolutions. The Cauchy foliation slices the worldtube into spherical cross-sections. The characteristic evolution is based upon the outgoing null hypersurfaces emanating from these slices, with the evolution proceeding from one hypersurface to the next by the outward radial march described earlier. There is no need to truncate spacetime at a finite distance from the sources, since compactification of the radial null coordinate makes it possible to cover the infinite space with a finite computational grid. In this way, the true waveform may be directly computed by a finite difference algorithm. Although characteristic evolution has limitations in regions where caustics develop, it proves to be both accurate and computationally efficient in the treatment of exterior regions.
CCM evolves a mixed spacelike-null initial value problem in
which Cauchy data is given in a spacelike region bounded by a
spherical boundary
and characteristic data is given on a null hypersurface
emanating from
. The general idea is not entirely new. An early mathematical
investigation combining space-like and characteristic
hypersurfaces appears in the work of Duff [55]. The three chief ingredients for computational implementation
are: (i) a Cauchy evolution module, (ii) a characteristic
evolution module and (iii) a module for matching the Cauchy and
characteristic regions across an interface. The interface is the
timelike worldtube which is traced out by the flow of
along the worldlines of the Cauchy evolution, as determined by
the choice of lapse and shift. Matching provides the exchange of
data across the worldtube to allow evolution without any further
boundary conditions, as would be necessary in either a purely
Cauchy or a purely characteristic evolution.
CCM may be formulated as a purely analytic approach, but its
advantages are paramount in the solution of nonlinear problems
where analytic solutions would be impossible. One of the first
applications of CCM was a hybrid numerical-analytical version,
initiated by Anderson and Hobill for the 1D wave equation [5] (see below). There the characteristic region was restricted to
the far field where it was handled analytically by a linear
approximation.
The full potential of CCM lies in a purely numerical treatment
of nonlinear systems where its error converges to zero in the
continuum limit of infinite grid resolution [24,
25,
45
]. For high accuracy, CCM is also by far the most efficient
method. For small target error
, it has been shown that the relative amount of computation
required for CCM (
) compared to that required for a pure Cauchy calculation (
) goes to zero,
as
[31,
28
]. An important factor here is the use of a compactified
characteristic evolution, so that the whole spacetime is
represented on a finite grid. From a numerical point of view this
means that the only error made in a calculation of the radiation
waveform at infinity is the controlled error due to the finite
discretization. Accuracy of a Cauchy algorithm which uses an ABC
requires a large grid domain in order to avoid error from
nonlinear effects in the exterior. The computational demands of
matching are small because the interface problem involves one
less dimension than the evolution problem. Because characteristic
evolution algorithms are more efficient than Cauchy algorithms,
the efficiency can be further enhanced by making the matching
radius as small as consistent with avoiding caustics.
At present, the purely computational version of CCM is
exclusively the tool of general relativists who are used to
dealing with novel coordinate systems. A discussion of its
potential appears in [24]. Only recently [45,
46
,
54
,
26
] has its practicability been carefully explored. Research on
this topic has been stimulated by the requirements of the Binary
Black Hole Grand Challenge Alliance, where CCM was one of the
strategies being pursued to provide the boundary conditions and
determine the radiation waveform. But I anticipate that its use
will eventually spread throughout computational physics because
of its inherent advantages in dealing with hyperbolic systems,
particularly in three-dimensional problems where efficiency is
desired. A detailed study of the stability and accuracy of CCM
for linear and non-linear wave equations has been presented in
Ref. [27
], illustrating its potential for a wide range of problems.
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Characteristic Evolution and Matching
Jeffrey Winicour http://www.livingreviews.org/lrr-2001-3 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |