### 4.7 Stable implementation of 3D linearized Cauchy-characteristic matching

Although the individual pieces of the CCM module have been calibrated to give a second order accurate
interface between Cauchy and characteristic evolution modules in 3D general relativity, its stability has not
yet been established [39]. However, a stable version of CCM for linearized gravitational theory has recently
been demonstrated [208]. The Cauchy evolution is carried out using a harmonic formulation for
which the reduced equations have a well-posed initial-boundary problem. Previous attempts at
CCM were plagued by boundary induced instabilities of the Cauchy code. Although stable
behavior of the Cauchy boundary is only a necessary and not a sufficient condition for CCM,
the tests with the linearized harmonic code matched to a linearized characteristic code were
successful.
The harmonic conditions consist of wave equations which can be used to propagate the gauge as four
scalar waves using characteristic evolution. This allows the extraction world tube to be placed at a finite
distance from the injection world tube without introducing a gauge ambiguity. Furthermore, the harmonic
gauge conditions are the only constraints on the Cauchy formalism so that gauge propagation also insures
constraint propagation. This allows the Cauchy data to be supplied in numerically benign Sommerfeld form,
without introducing constraint violation. Using random initial data, robust stability of the CCM algorithm
was confirmed for 2000 crossing times on a Cauchy grid. Figure 7 shows a sequence of profiles of
the metric component as a linearized wave propagates cleanly through the
spherical injection boundary and passes to the characteristic grid, where it is propagated to
.