Figure 1:
Image of the lines under the map , . 

Figure 2:
Regions of absolute stability of RK time integrators with . Secondorder RK (RK2) and the Euler method are not locally stable. 

Figure 3:
Comparison [278] between numerical solutions to (7.122) (left, and right labeled as Nonstrictly stable) and (7.124) (right, labeled as Strictly stable). The left panel shows that the scheme is numerically stable but not timestable. The right panel shows that a conservative scheme is timestable and considerably more accurate. Reprinted with permission from [278]; copyright by IOP. 

Figure 4:
Semidiscrete spectrum for the advection equation and freezing boundary conditions [Eqs. (10.1, 10.2, 10.3], a Chebyshev collocation method, and numerical injection of the boundary condition. The number of collocation points is (from left to right) . The scheme passes the von Neumann condition (no positive real component in the spectrum). In fact, as discussed in the body of the text, the method can be shown to actually be numerically stable. 

Figure 5:
Spectrum of the Chebyshev penalty method for the advection equation and (from left to right) collocation points and . 

Figure 6:
Maximum real component (left) and spectral radii (right) versus penalty strength , for the Chebyshev penalty method for the advection equation with two domains ( collocation points). 

Figure 7:
Comparison of different numerical boundary approaches for an advectiondiffusion equation where the initial data is perturbed, introducing an inconsistency with the boundary condition [296]. The SAT approach, besides guaranteeing time stability for general systems, “washes out” this inconsistency in time. “MODIFIED P” corresponds to a modified projection [226] for which this is solved at the expense of losing an energy estimate. Courtesy: Ken Mattsson. Reprinted with permission from [296]; copyright by Springer. 

Figure 8:
Comparison between boundary conditions in case (a) (solid) and case (b) (dotted); see the body of the text for more details. Four different resolutions are shown: , , and , where and refer to the number of collocation points in the radial and angular directions, where Chebyshev and spherical harmonics are used, respectively. Left panel: the difference between the solution with outer boundary at and the reference solution. Right panel: the constraint violation (see [366] for precise definitions of these quantities and further details). Courtesy: Oliver Rinne. Reprinted with permission from [366]; copyright by IOP. 

Figure 9:
Comparison of the time Fourier transform of and for two different radii of the outer boundary. The leveling off of for is due to numerical roundoff effects (note the magnitude of at those frequencies). Courtesy: Oliver Rinne. Reprinted with permission from [366]; copyright by IOP. 

Figure 10:
Early (left) and late (right) stages of the apparent horizon describing the evolution of an unstable black string. Courtesy: Luis Lehner and Frans Pretorius. 

Figure 11:
Left: Legosphere around a blackhole binary. Right: Mesh refinement around two black holes. Courtesy: Vitor Cardoso, Ulrich Sperhake and Helvi Witek. Reprinted with permission from [438]; copyright by APS. 

Figure 12:
Combining adaptive mesh refinement with curvilinear grids adapted to the wave zone. Courtesy: Denis Pollney. Reprinted with permission: top from [332], bottom from [334]; copyright by APS. 

Figure 13:
Sample domain decomposition used in spectral evolutions of blackhole binaries. The bottom plot illustrates how the coordinate shape of the excision domain is kept proportional to the coordinate shape of the black holes themselves. Courtesy: Bela Szilágyi. 

Figure 14:
Domain decomposition used in evolutions of accretion disks around black holes [256]; see the text for more details. Courtesy: Oleg Korobkin. 

Figure 15:
Equatorial cut of the computational domain used in multiblock simulations of orbiting blackhole binaries (left). Schematic figure showing the direction considered as radial (red arrows) for the cuboidal blocks (right). Reprinted with permission from [326]; copyright by APS. 

Figure 16:
Multiblock domain decomposition for a binary blackhole simulation. Reprinted with permission from [326]; copyright by APS. 

Figure 17:
Equatorial crosssection of varations of cubed sphere patches. Reprinted with permission from [326]; copyright by APS. 
http://www.livingreviews.org/lrr20129 
Living Rev. Relativity 15, (2012), 9
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