### 5.8 Stable 3D linearized Cauchy-characteristic matching

Although the individual pieces of the CCM module have been calibrated to give an accurate interface
between Cauchy and characteristic evolution modules in 3D general relativity, its stability has not yet been
established [52]. However, a stable version of CCM for linearized gravitational theory has recently been
demonstrated [287]. 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 for the coordinates, which can be used to
propagate the gauge as four scalar waves using characteristic evolution. This allows the extraction
worldtube to be placed at a finite distance from the injection worldtube 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 45^{3} 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
.