5.8 Stable 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 [45]. However, a stable version of CCM for linearized gravitational theory has recently been demonstrated [236]. 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 453 Cauchy grid. Figure 7View Image shows a sequence of profiles of the metric component xy √ --- xy γ = − gg as a linearized wave propagates cleanly through the spherical injection boundary and passes to the characteristic grid, where it is propagated to ℐ+.

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Figure 7: Sequence of slices of the metric component γxy, evolved with the linear matched Cauchy-characteristic code. In the last snapshot, the wave has propagated cleanly onto the characteristic grid with negligible remnant noise.

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