In order to numerically solve Einstein's field equations it is necessary to cast the equations in a form suitable for a computer based treatment. Among the formulations proposed for this purpose by far the most frequently applied is the canonical "3+1" decomposition of Arnowitt, Deser and Misner (1962) commonly referred to as the ADM formalism. In this approach spacetime is decomposed into a 1-parameter family of 3-dimensional space-like hypersurfaces and the Einstein equations are put into the form of an initial value problem. Initial data is provided on one hypersurface in the form of the spatial 3-metric and its time derivative and this data is evolved subject to certain constraints and the specification of gauge choices.
It is a known problem, however, that the ADM formalism does not result in a strictly hyperbolic formulation of the Einstein equations and in combination with its complicated structure the stability properties of the ensuing finite differencing schemes remain unclear. These difficulties have given rise to the development of modified versions of the ADM formulation in which the Einstein equations are written in "somewhat more hyperbolic". Such modifications of the canonical ADM scheme have been successfully tested, but "the" optimal "3+1" formulation has yet to be found.
An entirely different approach to the field equations is based on the decomposition of spacetime into families of null-surfaces, the characteristic surfaces of the propagation of gravitational radiation. The Einstein field equations are again formulated as an initial value problem and by virtue of a suitable choice of characteristic coordinates one obtains a natural classification of the equations into evolution and hypersurface equations. The characteristic initial value problem was first formulated by Bondi et al. (1962) and Sachs (1962) in order to facilitate a rigorous analysis of gravitational radiation which is properly described at null infinity only. It is a generic drawback of "3+1" formulations that null infinity cannot be included in the numerical grid by means of compactifying spacetime and instead outgoing radiation boundary conditions need to be used at finite radius. Aside from the non-rigorous analysis of gravitational radiation at finite distances these artificial boundary conditions give rise to spurious numerical reflections. A characteristic formulation resolves these problems in a natural way but is itself vulnerable to the formation of caustics in regions of strong curvature.
It is these properties of "3+1" formulations and the characteristic method that resulted in the idea of Cauchy characteristic matching (CCM), i.e. the combination of a "3+1" scheme applied in the interior and a characteristic formalism in the outer vacuum region. This allows one to make use of the advantages of both methods.