List of Figures

View Image Figure 1:
Image of the lines Re (z) = const > 0 under the map ℂ → ℂ, √ -2---- z ↦→ z + z + 1.
View Image Figure 2:
Regions of absolute stability of RK time integrators with s = n. Second-order RK (RK2) and the Euler method are not locally stable.
View Image Figure 3:
Comparison [278] between numerical solutions to (7.122View Equation) (left, and right labeled as Non-strictly stable) and (7.124View Equation) (right, labeled as Strictly stable). The left panel shows that the scheme is numerically stable but not time-stable. The right panel shows that a conservative scheme is time-stable and considerably more accurate. Reprinted with permission from [278]; copyright by IOP.
View Image Figure 4:
Semi-discrete spectrum for the advection equation and freezing boundary conditions [Eqs. (10.1View Equation, 10.2View Equation, 10.3View Equation], a Chebyshev collocation method, and numerical injection of the boundary condition. The number of collocation points is (from left to right) N = 20,40,60. 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.
View Image Figure 5:
Spectrum of the Chebyshev penalty method for the advection equation and (from left to right) N = 20,40,60 collocation points and S = 0.3.
View Image Figure 6:
Maximum real component (left) and spectral radii (right) versus penalty strength S, for the Chebyshev penalty method for the advection equation with two domains (N = 20 collocation points).
View Image Figure 7:
Comparison of different numerical boundary approaches for an advection-diffusion 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.
View Image 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: (Nr, L) = (21,8), (31,10), (41,12 ) and (51,14), where N r and L 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 Δ 𝒰 between the solution with outer boundary at R = 41.9M and the reference solution. Right panel: the constraint violation 𝒞 (see [366] for precise definitions of these quantities and further details). Courtesy: Oliver Rinne. Reprinted with permission from [366]; copyright by IOP.
View Image Figure 9:
Comparison of the time Fourier transform of Ψ0 and Ψ4 for two different radii of the outer boundary. The leveling off of Ψ0 for kM ≳ 3 is due to numerical roundoff effects (note the magnitude of Ψ 0 at those frequencies). Courtesy: Oliver Rinne. Reprinted with permission from [366]; copyright by IOP.
View Image 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.
View Image Figure 11:
Left: Lego-sphere around a black-hole binary. Right: Mesh refinement around two black holes. Courtesy: Vitor Cardoso, Ulrich Sperhake and Helvi Witek. Reprinted with permission from [438]; copyright by APS.
View Image 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.
View Image Figure 13:
Sample domain decomposition used in spectral evolutions of black-hole 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.
View Image Figure 14:
Domain decomposition used in evolutions of accretion disks around black holes [256]; see the text for more details. Courtesy: Oleg Korobkin.
View Image Figure 15:
Equatorial cut of the computational domain used in multi-block simulations of orbiting black-hole 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.
View Image Figure 16:
Multi-block domain decomposition for a binary black-hole simulation. Reprinted with permission from [326]; copyright by APS.
View Image Figure 17:
Equatorial cross-section of varations of cubed sphere patches. Reprinted with permission from [326]; copyright by APS.