4.7 3D Cauchy-Characteristic Matching4 Cauchy-Characteristic Matching4.5 Axisymmetric Cauchy-Characteristic Matching

4.6 Cauchy-Characteristic Matching for 3D Scalar Waves

CCM has been successfully implemented in the fully 3D problem of nonlinear scalar waves evolving in a flat spacetime [27, 26]. This study demonstrated the feasibility of matching between Cartesian Cauchy coordinates and spherical null coordinates, the setup required to apply CCM to the binary black hole problem. Unlike the previous examples of matching, the Cauchy and characteristic patches now did not share a common coordinate which can be used to define the matching interface. This introduced a major complication into the matching procedure, resulting in extensive use of inter-grid interpolation. The accompanying short wavelength numerical noise presented a new challenge in obtaining a stable algorithm.

The nonlinear waves were modeled by the equation


with self-coupling tex2html_wrap_inline2337 and external source S . The initial Cauchy data tex2html_wrap_inline2341 and tex2html_wrap_inline2343 are assigned in a spatial region bounded by a spherical matching surface of radius tex2html_wrap_inline2345 .

The characteristic initial value problem (21Popup Equation) is expressed in standard spherical coordinates tex2html_wrap_inline2347 and retarded time tex2html_wrap_inline2349 :


where tex2html_wrap_inline1731 and tex2html_wrap_inline2353 is the angular momentum operator


The initial null data consist of tex2html_wrap_inline2355 on the outgoing characteristic cone tex2html_wrap_inline2357 emanating from the matching worldtube at the initial Cauchy time.

CCM was implemented so that, in the continuum limit, tex2html_wrap_inline1975 and its normal derivatives would be continuous across the interface tex2html_wrap_inline2361 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.

4.7 3D Cauchy-Characteristic Matching4 Cauchy-Characteristic Matching4.5 Axisymmetric Cauchy-Characteristic Matching

image Characteristic Evolution and Matching
Jeffrey Winicour
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de