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,

where is the state vector, are complex matrices, , and is a complex matrix. As before, we assume for simplicity that all coefficients belong to the class of bounded, smooth functions with bounded derivatives. The data consists of the initial data and the boundary data .Compared to the initial-value problem discussed in Section 3 the following new issues and difficulties appear when boundaries are present:

- For a smooth solution to exist, the data and must satisfy appropriate compatibility conditions at the intersection between the initial and boundary surface [344]. Assuming that is continuous, for instance, Eqs. (5.2, 5.3) imply that for all . If is continuously differentiable, then taking a time derivative of Eq. (5.3) and using Eqs. (5.1, 5.2) leads to where is the complex matrix with coefficients Assuming higher regularity of , one obtains additional compatibility conditions by taking further time derivatives of Eq. (5.3). In particular, for an infinitely-differentiable solution , one has an infinite family of such compatibility conditions at , and one must make sure that the data , satisfies each of them if the solution is to be reproduced by the IBVP. If an exact solution of the partial-differential equation (5.1) is known, a convenient way of satisfying these conditions is to choose the data such that in a neighborhood of , and agree with the corresponding values for , i.e., such that and for in a neighborhood of . However, depending on the problem at hand, this might be too restrictive.
- The next issue is the question of what class of boundary conditions (5.3) leads to a well-posed problem. In particular, one would like to know, which are the restrictions on the matrix implying existence of a unique solution, provided the compatibility conditions hold. In order to illustrate this issue on a very simple example, consider the advection equation on the interval . The most general solution has the form for some differentiable function . The function is determined on the interval by the initial data alone, and so the initial data alone fixes the solution on the strip . Therefore, one is not allowed to specify any boundary conditions at , whereas data must be specified for at in order to uniquely determine the function on the interval .
- Additional difficulties appear when the system has constraints, like in the case of electromagnetism and general relativity. In the previous Section 4, we saw in the case of Einstein’s equations that it is usually sufficient to solve these constraints on an initial Cauchy surface, since the Bianchi identities and the evolution equations imply that the constraints propagate. However, in the presence of boundaries one can only guarantee that the constraints remain satisfied inside the future domain of dependence of the initial surface unless the boundary conditions are chosen with care. Methods for constructing constraint-preserving boundary conditions, which make sure that the constraints propagate correctly on the whole spacetime domain will be discussed in Section 6.

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.

5.1 The Laplace method

5.1.1 Necessary conditions for well-posedness and the Lopatinsky condition

5.1.2 Sufficient conditions for well-posedness and boundary stability

5.1.3 Second-order systems

5.2 Maximal dissipative boundary conditions

5.2.1 Application to systems of wave equations

5.2.2 Existence of weak solutions and the adjoint problem

5.3 Absorbing boundary conditions

5.3.1 The one-dimensional wave equation

5.3.2 The three-dimensional wave equation

5.3.3 The wave equation on a curved background

5.1.1 Necessary conditions for well-posedness and the Lopatinsky condition

5.1.2 Sufficient conditions for well-posedness and boundary stability

5.1.3 Second-order systems

5.2 Maximal dissipative boundary conditions

5.2.1 Application to systems of wave equations

5.2.2 Existence of weak solutions and the adjoint problem

5.3 Absorbing boundary conditions

5.3.1 The one-dimensional wave equation

5.3.2 The three-dimensional wave equation

5.3.3 The wave equation on a curved background

Living Rev. Relativity 15, (2012), 9
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