11.5 Multi-domain studies of accretion disks around black holes

The freedom to choose arbitrary (smooth) coordinate transformations allows the design of sophisticated problem-fitted meshes to address a number of practical issues. In [256Jump To The Next Citation Point] the authors used a hybrid multiblock approach for a general-relativistic hydrodynamics code developed in [456Jump To The Next Citation Point] to study instabilities in accretion disks around black holes in the context of gamma-ray–burst central engines. They evolved the spacetime metric using the first-order form of the generalized harmonic formulation of the Einstein equations (see Section 4.1) on conforming grids, while using a high-resolution shock capturing scheme for relativistic fluids on the same grid but with additional overlapping boundary zones (see [456] for details on the method). The metric differentiation was performed using the optimized D8 −4 FD operators satisfying the SBP property, as described in Section 8.3. The authors made extensive use of adapted curvilinear coordinates in order to achieve desired resolutions in different parts of the domain and to make the coordinate lines conform to the shape of the solution. Maximal dissipative boundary conditions as defined in Section 5.2 were applied to the incoming fields, and inter-domain boundary conditions for the metric were implemented using the finite-difference version of the penalty method described in Section 10.

Figure 14View Image shows examples of the type of mesh adaptation used. The top left panel shows the meridional cut of an accretion disk on a uniform multiblock mesh (model C in [256Jump To The Next Citation Point]). The top right panel gives an example of a mesh with adapted radial coordinate lines, which resolves the disk more accurately than the mesh with uniform grid resolution (see [256Jump To The Next Citation Point] for details on the particular coordinate transformations used to obtain such a grid). A 3D view of such multi-domain mesh at large radii is shown on the bottom left panel of Figure 14View Image. In the area near the inner radius where the central black hole is located, the resolution is high enough to accurately resolve the shape and the dynamics of the black-hole horizon. Finally, near the disk, the resolution across the disk in both radial and angular directions is made approximately equal and sufficiently high to resolve the transverse disk dynamics. The bottom right panel of Figure 14View Image shows the 3D view of the adapted mesh in the vicinity of the disk.

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.

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