The initial work on 3D characteristic evolution led to two independent codes, one developed at Canberra and the other at Pittsburgh (the PITT code), both with the capability to study gravitational waves in single black hole spacetimes at a level not yet mastered at the time by Cauchy codes. The Pittsburgh group established robust stability and second order accuracy of a fully nonlinear 3D code which was able to calculate the waveform at null infinity [49, 46] and to track a dynamical black hole and excise its internal singularity from the computational grid [119, 116]. The Canberra group implemented an independent nonlinear 3D code which accurately evolved the exterior region of a Schwarzschild black hole. Both codes pose data on an initial null hypersurface and on a worldtube boundary, and evolve the exterior spacetime out to a compactified version of null infinity, where the waveform is computed. However, there are essential differences in the underlying geometrical formalisms and numerical techniques used in the two codes and in their success in evolving generic black hole spacetimes. Recently two new codes have evolved from the PITT code by introducing a new choice of spherical coordinates [111, 201].

4.1 Coordinatization of the sphere

4.1.1 Stereographic grids

4.1.2 Cubed sphere grids

4.1.3 Toroidal grids

4.2 Geometrical formalism

4.2.1 Angular dissipation

4.2.2 First versus second differential order

4.2.3 Numerical methods

4.2.4 Stability

4.2.5 Accuracy

4.2.6 Nonlinear scattering off a Schwarzschild black hole

4.2.7 Black hole in a box

4.3 Characteristic treatment of binary black holes

4.4 Perturbations of Schwarzschild

4.4.1 Close approximation white hole and black hole waveforms

4.4.2 Fissioning white hole

4.5 Nonlinear mode coupling

4.6 3D Einstein–Klein–Gordon system

4.1.1 Stereographic grids

4.1.2 Cubed sphere grids

4.1.3 Toroidal grids

4.2 Geometrical formalism

4.2.1 Angular dissipation

4.2.2 First versus second differential order

4.2.3 Numerical methods

4.2.4 Stability

4.2.5 Accuracy

4.2.6 Nonlinear scattering off a Schwarzschild black hole

4.2.7 Black hole in a box

4.3 Characteristic treatment of binary black holes

4.4 Perturbations of Schwarzschild

4.4.1 Close approximation white hole and black hole waveforms

4.4.2 Fissioning white hole

4.5 Nonlinear mode coupling

4.6 3D Einstein–Klein–Gordon system

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