5 Boundary Conditions: The Initial-Boundary Value Problem

In Section 3 we discussed the general Cauchy problem for quasilinear hyperbolic evolution equations on the unbounded domain ℝn. 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 Σ ⊂ ℝn with C∞-smooth boundary ∂Σ. Let T > 0. We are considering an IBVP of the following form,

n u = ∑ Aj(t,x,u) ∂-u + F (t,x,u), x ∈ Σ, t ∈ [0,T], (5.1 ) t j=1 ∂xj u(0,x) = f(x), x ∈ Σ, (5.2 ) b(t,x,u)u = g(t,x ), x ∈ ∂Σ, t ∈ [0,T ], (5.3 )
where m u(t,x) ∈ ℂ is the state vector, 1 n A (t,x,u),...,A (t,x,u) are complex m × m matrices, m F (t,x,u) ∈ ℂ, and b(t,x,u) is a complex r × m matrix. As before, we assume for simplicity that all coefficients belong to the class C∞b ([0,T] × Σ × ℂm ) of bounded, smooth functions with bounded derivatives. The data consists of the initial data f ∈ C∞b (Σ, ℂm ) and the boundary data g ∈ C ∞ ([0,T ] × ∂ Σ,ℂr ) b.

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 [258Jump To The Next Citation Point] 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 [241Jump To The Next Citation Point].

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 (5.1View Equation) 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

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