Cauchy-characteristic matching for scalar waves

The combination of Cauchy (3+1) and characteristic (2+2) evolution algorithms is a new approach for the accurate numerical solution of initial-value problems involving wave emission from isolated sources. In this approach, a conventional Cauchy initial-value formulation is used in an bounded interior region including the sources, and a formulation based on characteristic hypersurfaces is used to evolve the radiation field in the whole exterior region (including infinity). Special numerical procedures are used to transfer information across the interface between the two computational regions, in a way that must be consistent with the assumed smoothness of the fields at the interface. These numerical procedures replace the artificial boundary conditions which are usually employed in conjunction with Cauchy evolution codes in unbounded regions.

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:

The generalized Sommerfeld approach to scalar CCM has been implemented in the software package Cc, whose early versions (August 1994) were the first succesful implementations of CCM in three spatial dimensions.

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