In Section 3 we discussed the general Cauchy problem for quasilinear hyperbolic evolution equations on the
unbounded domain . However, in the numerical modeling of such problems one is faced with
the finiteness of computer resources. A common approach for dealing with this problem is to
truncate the domain via an artificial boundary, thus forming a finite computational domain
with outer boundary. Absorbing boundary conditions must then be specified at the boundary
such that the resulting IBVP is well posed and such that the amount of spurious reflection is
minimized.
Therefore, we examine in this section quasilinear hyperbolic evolution equations on a finite, open domain
with
-smooth boundary
. Let
. We are considering an IBVP of the following
form,
Compared to the initial-value problem discussed in Section 3 the following new issues and difficulties appear when boundaries are present:
There are two common techniques for analyzing an IBVP. The first, discussed in Section 5.1, is based on
the linearization and localization principles, and reduces the problem to linear, constant coefficient IBVPs
which can be explicitly solved using Fourier transformations, similar to the case without boundaries. This
approach, called the Laplace method, is very useful for finding necessary conditions for the well-posedness of
linear, constant coefficient IBVPs. Likely, these conditions are also necessary for the quasilinear IBVP, since
small-amplitude high-frequency perturbations are essentially governed by the corresponding linearized,
frozen coefficient problem. Based on the Kreiss symmetrizer construction [258] and the theory of
pseudo-differential operators, the Laplace method also gives sufficient conditions for the linear,
variable coefficient problem to be well posed; however, the general theory is rather technical.
For a discussion and interpretation of this approach in terms of wave propagation we refer
to [241
].
The second method, which is discussed in Section 5.2, is based on energy inequalities obtained from
integration by parts and does not require the use of pseudo-differential operators. It provides a class of
boundary conditions, called maximal dissipative, which leads to a well-posed IBVP. Essentially, these
boundary conditions specify data to the incoming normal characteristic fields, or to an appropriate
linear combination of the in- and outgoing normal characteristic fields. Although technically less
involved than the Laplace one, this method requires the evolution equations (5.1) to be symmetric
hyperbolic in order to be applicable, and it gives sufficient, but not necessary, conditions for
well-posedness.
In Section 5.3 we also discuss absorbing boundary conditions, which are designed to minimize spurious reflections from the boundary surface.
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Living Rev. Relativity 15, (2012), 9
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