11.3 Adaptive mesh refinement and curvilinear grids

In  [331Jump To The Next Citation Point, 333] the authors introduced an approach, which combines the advantages of adaptive mesh refinement near the “sources” (say, black holes) with curvilinear coordinates adapted to the wave zone; see Figure 12View Image. The patches are communicated using polynomial interpolation in its Lagrange form, as explained in Section 8.1, and centered stencils are used, both for finite differencing and interpolation. Up to eighth-order finite differencing is used, with an observed convergence rate between six and eight in the (ℓ = 2 = m ) modes of the computed gravitational waves (parts of the scheme have a lower-order convergence rate, but they do not appear to dominate). Presently, the BSSN formulation of Einstein’s equations as described in Section 4.3 is used directly in its second-order-in-space form, with outgoing boundary conditions for all the fields. The implementation is generic and flexible enough to allow for other systems of equations, though. As in most approaches using curvilinear coordinates in numerical relativity, the field variables are expressed in a global coordinate frame. This might sound unnatural and against the idea of using local patches and coordinates. However, it simplifies dramatically any implementation. It is also particularly important when taking into account that most formulations of Einstein’s equations and coordinate conditions used are not covariant.

This hybrid approach has been used in several applications, including the validation of extrapolation procedures of gravitational waves extracted from numerical simulations at finite radii to large distances from the “sources” [331]. Since the outermost grid structure is well adapted to the wave zone, the outer boundary can be located at large distances with only linear cost on its location. Other applications include Cauchy-Characteristic extraction (CCE) of gravitational waves [350, 55], a waveform hybrid development [371], and studies of memory effect in gravitational waves [330]. The accuracy necessary to study small memory effects is enabled both by the grid structure – being able to locate the outer boundary far away – and CCE.

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.

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