The nonlinear waves were modeled by the equation
with self-coupling and external source S . The initial Cauchy data and are assigned in a spatial region bounded by a spherical matching surface of radius .
The characteristic initial value problem (21) is expressed in standard spherical coordinates and retarded time :
where and is the angular momentum operator
The initial null data consist of on the outgoing characteristic cone emanating from the matching worldtube at the initial Cauchy time.
CCM was implemented so that, in the continuum limit, and its normal derivatives would be continuous across the interface between the regions of Cauchy and characteristic evolution. The use of a Cartesian discretization in the interior and a spherical discretization in the exterior complicated the treatment of the interface. In particular, the stability of the matching algorithm required careful attention to the details of the inter-grid matching. Nevertheless, there was a reasonably broad range of discretization parameters for which CCM was stable.
Two different ways of handling the spherical coordinates were used. One was based upon two overlapping stereographic grid patches and the other upon a multiquadric approximation using a quasi-regular triangulation of the sphere. Both methods gave similar accuracy. The multiquadric method showed a slightly larger range of stability. Also, two separate tactics were used to implement matching, one based upon straightforward interpolations and the other upon maintaining continuity of derivatives in the outward null direction (a generalization of the Sommerfeld condition). Both methods were stable for a reasonable range of grid parameters. The solutions were second order accurate and the Richardson extrapolation technique could be used to accelerate convergence.
The performance of CCM was compared to traditional ABC's. As expected, the nonlocal ABC's yielded convergent results only in linear problems, and convergence was not observed for local ABC's, whose restrictive assumptions were violated in all of the numerical experiments. The computational cost of CCM was much lower than that of current nonlocal conditions. In strongly nonlinear problems, matching appears to be the only available method which is able to produce numerical solutions which converge to the exact solution with a fixed boundary.
|Characteristic Evolution and Matching
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