5.7 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 [50, 49]. 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 spherically or cylindrically symmetric examples of matching, the Cauchy and characteristic patches do not 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 intergrid interpolation. The accompanying short wavelength numerical noise presents a challenge in obtaining a stable algorithm.

The nonlinear waves were modeled by the equation

c−2∂2tΦ = ∇2Φ + F (Φ ) + S(t,x,y,z), (66 )
with self-coupling F (Φ ) and external source S. The initial Cauchy data Φ(t0,x,y,z ) and ∂tΦ (t0,x, y,z) are assigned in a spatial region bounded by a spherical matching surface of radius Rm.

The characteristic initial value problem (66View Equation) is expressed in standard spherical coordinates (r,𝜃,φ ) and retarded time u = t − r + R m:

2 L2g 2∂u∂rg = ∂ rg − --2-+ r(F + S), (67 ) r
where g = rΦ and 2 L is the angular momentum operator
2 L2g = − ∂𝜃(sin-𝜃∂-𝜃g)− -∂φg-. (68 ) sin 𝜃 sin2 𝜃
The initial null data consist of g(r,𝜃,φ,u0) on the outgoing characteristic cone u0 = t0 emanating at the initial Cauchy time from the matching worldtube at r = Rm.

CCM was implemented so that, in the continuum limit, Φ and its normal derivatives would be continuous across the matching interface. 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 intergrid 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 ABCs. As expected, the nonlocal ABCs yielded convergent results only in linear problems, and convergence was not observed for local ABCs, whose restrictive assumptions were violated in all of the numerical experiments. The computational cost of CCM was much lower than that of current nonlocal boundary conditions. In strongly nonlinear problems, CCM appears to be the only available method, which is able to produce numerical solutions that converge to the exact solution with a fixed boundary.

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