Cauchy–Characteristic Evolution and Waveforms

Abstract

We investigate a new methodology for the computation of waves generated by isolated sources. This approach consists of a global spacetime evolution algorithm based on a Cauchy initial-value formulation in a bounded interior region and based on characteristic hypersurfaces in the exterior; we match the two schemes at their common interface. The characteristic formulation allows accurate description of radiative infinity in a compactified finite coordinate interval, so that our numerical solution extends to infinity and accurately models the free-space problem. The matching interface need not be situated far from the sources, the wavefronts may have arbitrary nonspherical geometry, and strong nonlinearity may be present in both the interior and the exterior regions. Stability and second-order convergence of the algorithms (to the exact solution of the infinite-domain problem) are established numerically in three space dimensions. The matching algorithm is compared with examples of both local and nonlocal radiation boundary conditions proposed in the literature. For linear problems, matching outperformed the local radiation conditions chosen for testing, and was about as accurate (for the same grid resolution) as the exact nonlocal conditions. However, since the computational cost of the nonlocal conditions is many times that of matching, this algorithm may be used with higher grid resolutions, yielding a significantly higher final accuracy. For strongly nonlinear problems, matching was significantly more accurate than all other methods tested. This seems to be due to the fact that currently available local and nonlocal conditions are based on linearizing the governing equations in the far field, while matching consistently takes nonlinearity into account in both interior and exterior regions.