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Figure 1:
Image of the lines ![]() ![]() ![]() |
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Figure 2:
Regions of absolute stability of RK time integrators with ![]() |
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Figure 3:
Comparison [278] between numerical solutions to (7.122 ![]() ![]() |
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Figure 4:
Semi-discrete spectrum for the advection equation and freezing boundary conditions [Eqs. (10.1 ![]() ![]() ![]() ![]() |
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Figure 5:
Spectrum of the Chebyshev penalty method for the advection equation and (from left to right) ![]() ![]() |
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Figure 6:
Maximum real component (left) and spectral radii (right) versus penalty strength ![]() ![]() |
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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. |
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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: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Figure 9:
Comparison of the time Fourier transform of ![]() ![]() ![]() ![]() ![]() |
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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. |
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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. |
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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. |
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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. |
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Figure 14:
Domain decomposition used in evolutions of accretion disks around black holes [256]; see the text for more details. Courtesy: Oleg Korobkin. |
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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. |
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Figure 16:
Multi-block domain decomposition for a binary black-hole simulation. Reprinted with permission from [326]; copyright by APS. |
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Figure 17:
Equatorial cross-section of varations of cubed sphere patches. Reprinted with permission from [326]; copyright by APS. |
http://www.livingreviews.org/lrr-2012-9 |
Living Rev. Relativity 15, (2012), 9
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