The von Neumann stability analysis of the interior algorithm linearizes the equations, while
assuming a uniform grid with periodic boundary conditions, 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. The mode , with
real,
is not included in the von Neumann analysis for periodic boundary conditions. However, for
the half plane problem in the domain
, one can legitimately prescribe such a mode as
initial data as long as
so that it has finite energy. Thus the stability of such boundary
modes must be checked. In the case of an additional boundary, e.g. for a problem in the domain
, the Godunov–Ryaben’kii theory gives as a necessary condition for stability the
combined von Neumann stability of the interior and the stability of the allowed boundary
modes [224]. The Kreiss condition [132] strengthens this result by providing a sufficient condition for
stability.
The correct physical formulation of any Cauchy problem for an isolated system also involves asymptotic
conditions at infinity. These conditions must ensure not only that the total energy and energy loss by
radiation are both finite, but they must also ensure the proper asymptotic falloff of the radiation
fields. However, when treating radiative systems computationally, an outer boundary is often
established artificially at some large but finite distance in the wave zone, i.e. many wavelengths
from the source. Imposing an appropriate 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. Gustaffson and Kreiss have shown in general that the construction of a nonreflecting
boundary condition for an isolated system requires knowledge of the solution in a neighborhood of
infinity[131].
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 scalar wave problem with field where the initial
radiation is confined to a compact region inside the boundary, the missing Cauchy data outside
the boundary would be
at the initial time
. 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 in the scalar field
case where
is appropriate Cauchy data outside the boundary at
, it would
still be a non-trivial evolution problem to correctly assign the associated boundary data for
.
It is common practice in computational physics to impose an 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 [168, 154
, 138
, 202
],
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 resulting error is
of an analytic origin, essentially independent of computational discretization. In general, a
systematic reduction of this error can only be achieved by moving the computational boundary to
larger and larger radii. This is computationally very expensive, especially for 3-dimensional
simulations.
A traditional ABC for the wave equation is the Sommerfeld condition. For a scalar field satisfying
the Minkowski space wave equation
A homogeneous Sommerfeld condition, i.e., is exact only in the spherically symmetric case. The
Sommerfeld boundary data
in the general case (36
) falls off as
, so that a homogeneous
Sommerfeld condition introduces an error, which is small only for large
. As an example, for the dipole
solution
Much work has been done on formulating boundary conditions, both exact and approximate, for
linear problems in situations that are not spherically symmetric. These boundary conditions
are given various names in the literature, e.g., absorbing or non-reflecting. A variety of ABCs
have been reported for linear problems. See the articles [107, 202, 245
, 209
, 43
] for general
discussions.
Local ABCs have been extensively applied to linear problems with varying
success [168, 89
, 35
, 244
, 138
, 52, 155]. Some of these conditions are local approximations to exact
integral representations of the solution in the exterior of the computational domain [89
], while others are
based on approximating the dispersion relation of the so-called one-way wave equations [168, 244].
Higdon [138] showed that this last approach is essentially equivalent to specifying a finite number
of angles of incidence for which the ABCs yield perfect transmission. Local ABCs have also
been derived for the linear wave equation by considering the asymptotic behavior of outgoing
solutions [35], thus generalizing the Sommerfeld outgoing radiation condition. Although this type of
ABC is relatively simple to implement and has a low computational cost, the final accuracy is
often limited because the assumptions made about the behavior of the waves are rarely met in
practice [107
, 245
].
The disadvantages of local ABCs have led some workers to consider exact nonlocal boundary conditions
based on integral representations of the infinite domain problem [243, 107, 245
]. Even for problems where
the Green’s function is known and easily computed, such approaches were initially dismissed as
impractical [89
]; however, the rapid increase in computer power has made it possible to implement exact
nonlocal ABCs for the linear wave equation and Maxwell’s equations in 3D [80
, 126]. If properly
implemented, this method can yield numerical solutions to a linear problem which converge to the exact
infinite domain problem in the continuum limit, while 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 [107
, 80, 245
].
The extension of ABCs 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
ABCs [245, 209]. 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 a spurious sheet source
on the boundary, which contaminates both the interior and the exterior evolutions. But the
fields and their normal derivatives constitute an overdetermined set of data for the boundary
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 potentials method, to eliminate back reflection at the
boundary [245].
There have been only a few applications of ABCs to strongly nonlinear problems [107]. Thompson [240
]
generalized a previous nonlinear ABC of Hedstrom [137] 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 [240, 107]. Hagstrom and Hariharan [133] 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 approach has not yet been
validated.
In order to reduce the level of approximation at the analytic level, an artificial boundary for a nonlinear problem must be placed sufficiently far from the strong-field region. This sharply increases the computational cost in multi-dimensional simulations [89]. There is no numerical method which converges (as the discretization is refined) to the infinite domain exact solution of a strongly nonlinear wave problem in multi-dimensions, while keeping the artificial boundary fixed. Attempts to use compactified Cauchy hypersurfaces which extend the domain to spatial infinity have failed because the phase of short wavelength radiation varies rapidly in spatial directions [154]. Characteristic evolution avoids this problem by approaching infinity along the phase fronts.
CCM is a strategy that eliminates this nonlinear source of error. In the simplest version of 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 in Section 3.1). There is no need to truncate spacetime at a finite distance from the source, since compactification of the radial null coordinate used in the characteristic evolution makes it possible to cover the infinite space with a finite computational grid. In this way, the true waveform may be computed up to discretization error by the finite difference algorithm.
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