### 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 [256] the authors used a
hybrid multiblock approach for a general-relativistic hydrodynamics code developed in [456] 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 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 14 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 [256]). 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 [256] 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 14. 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 14 shows the 3D view of the adapted mesh in the vicinity of the
disk.