4.6 The CCM Gravitational Module4 CCM4.4 Numerical Matching for 1D

4.5 CCM for 3D Scalar Waves

CCM has been successfully implemented in the fully 3D problem of nonlinear scalar waves evolving in a flat space-time. [75, 92] The main purpose of the study was to demonstrate the feasibility of matching between Cartesian Cauchy coordinates and spherical null coordinates. This is the setup required to apply CCM to the binary black hole problem. Unlike the previous examples of matching, the Cauchy and characteristic patches do not now share a common coordinate which can be used to define the matching interface. This introduces a major complication into the matching procedure, resulting in extensive use of inter-grid interpolation. The accompanying short wavelength numerical noise presents a new challenge in obtaining a stable algorithm.

The nonlinear waves were modeled on the equation


with self-coupling tex2html_wrap_inline1951 and external source S . The initial Cauchy data tex2html_wrap_inline1955 and tex2html_wrap_inline1957 are assigned in a spatial region bounded by a spherical matching surface of radius tex2html_wrap_inline1959 .

The characteristic initial value problem (21Popup Equation) is expressed in standard spherical coordinates tex2html_wrap_inline1961 and retarded time tex2html_wrap_inline1963 :


where tex2html_wrap_inline1551 and tex2html_wrap_inline1967 is the angular momentum operator


The initial null data is tex2html_wrap_inline1969, on the outgoing characteristic cone tex2html_wrap_inline1971 emanating from the matching worldtube at the initial Cauchy time.

CCM was implemented so that, in the continuum limit, tex2html_wrap_inline1763 and its normal derivatives would be continuous across the interface tex2html_wrap_inline1975 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.

4.6 The CCM Gravitational Module4 CCM4.4 Numerical Matching for 1D

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