Most three-dimensional codes solving the Einstein equations currently use several non-uniform grids/numerical domains. Adaptive mesh refinement (AMR) à la Berger & Oliger , where the computational domain is covered with a set of nested grids, usually taken to be Cartesian ones, is used by many efforts. See, for instance, [386, 338, 394, 277, 160, 24, 393, 38, 84, 109, 430, 442, 439, 157, 321]). Other approaches use multiple patches with curvilinear coordinates, or a combination of both. Typical simulations of Einstein’s equations do not fall into the category of complex geometries and usually require a fairly “simple” domain decomposition (in comparison to fully unstructured approaches in other fields).
Below we give a brief overview of some domain decomposition approaches. Our discussion is far from exhaustive, and only a few representative samples from the rich variety of efforts are mentioned. In the context of Cauchy evolutions, the use of multiple patches in numerical relativity was first advocated and pursued by Thornburg [417, 418].
Living Rev. Relativity 15, (2012), 9
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