10 Numerical Boundary Conditions

In most practical computations, one inevitably deals with an IBVP and numerical boundary conditions have to be imposed. Usually the boundary is artificial and, as discussed in Section 5.3, absorbing boundary conditions are imposed. In other cases the boundary of the computational domain may actually represent infinity via compactification; see Section 6.4. Here we discuss some approaches for imposing numerical boundary conditions, with emphasis on sufficient conditions for stability based on the energy method, simplicity, and applicability to high order and spectral methods. In addition to outer boundaries, we also discuss interface ones appearing when there are multiple grids.

General stability results through the energy method are available for symmetric hyperbolic first-order linear systems with maximal dissipative boundary conditions. Unfortunately, in many cases of physical interest the boundary conditions are often neither in maximal dissipative form nor is the system linear. In particular, this is true for Einstein’s field equations, which are nonlinear, and, as we have seen in Section 6, require constraint-preserving absorbing boundary conditions, which do not always result in algebraic conditions on the fields at the boundary. Therefore, in many cases one does “the best that one can”, implementing the outer boundary conditions using discretizations, which are known to be stable, at least in the linearized, maximal dissipative case. Fortunately, since the outer boundaries are usually placed in the weak field, wave zone, more often than not this approach works well in practice. At the same time, it should be noted that the IBVPs for general relativity formulated in [187Jump To The Next Citation Point] and [264Jump To The Next Citation Point] (discussed in Section 5) are actually based on a symmetric hyperbolic first-order reduction of Einstein’s field equations with maximal dissipative boundary conditions (including constraint-preserving ones). Therefore, it should be possible to construct numerical schemes, which can be provably stable, at least in the linearized regime, using the techniques described in the last two Sections 8 and 9, and in Section 10.1 below. A numerical implementation of the formulations of [187] and [264] has not yet been pursued.

The situation at interface boundaries between grids, which are at least partially contained in the strong field region, is more subtle. Fortunately, only the characteristic structure of the equations is in principle needed at such boundaries, and not constraint-preserving boundary conditions. Methods for dealing with interfaces are discussed in Section 10.2.

Finally, in Section 10.3 we give an overview of some applications to numerical relativity of the boundary treatments discussed in Sections 10.1 and 10.2. As mentioned above, most of the techniques that we discuss have been mainly developed for first-order symmetric hyperbolic systems with maximal dissipative boundary conditions. In Section 10.3 we also point out ongoing and prospective work for second-order systems, as well as the important topic of absorbing boundary conditions in general relativity.

Most of the methods reviewed below involve decomposition of the principal part, its time derivative, or both, into characteristic variables, imposing the boundary conditions and changing back to the original variables. This can be done a priori, analytically, and the actual online numerical computational cost of these operations is negligible.

 10.1 Outer boundary conditions
  10.1.1 Injection
  10.1.2 Projections
  10.1.3 Penalty conditions
 10.2 Interface boundary conditions
  10.2.1 Penalty conditions
 10.3 Going further, applications in numerical relativity
  10.3.1 Absorbing boundary conditions

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