The main requirement for the validity of Cauchy-characteristic matching (CCM) is that the wave propagation be asymptotically nondispersive far from the sources. Both linear and nonlinear wave problems may be solved consistently by CCM, with relatively mild restrictions on the behavior of the nonlinear terms.

Although the main goal of CCM is to serve as an algorithm for the computation of gravitational radiation in General Relativity, it has been instructive to construct CCM algorithms for solving the scalar wave equation with a nonlinear term in flat space-time. This problem has some (but not all) of the essential ingredients of matching in General Relativity.

Among the many possible ways to carry out CCM, one that has proved to be
very accurate and
stable is based on exact generalizations of the usual Sommerfeld radiation boundary
condition. Such *generalized Sommerfeld conditions* may also be regarded as
continuity conditions for the normal derivative of the field across the matching
interface. The numerical treatment of the generalized conditions is similar to that
of the Sommerfeld condition, and may be carried out without introducing numerical
instabilities for a wide range of discretization parameters. Its accuracy has
been demonstrated for linear and nonlinear initial-value
problems with and without spherical symmetry in a series of systematic
numerical experiments, described in two papers:

- Cauchy-characteristic matching: a new approach to radiation boundary
conditions.
*Phys. Rev. Lett.***76**:4303 (1996). - Cauchy-characteristic evolution and waveforms,
submitted to
*J. Comp. Phys.*, 1996 (PostScript version available here ).

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