  Although the traditional form of the Einstein field equations is
intrinsically four-dimensional, the field of numerical general relativity
focuses on the reformulation of these equations as a 3 + 1-dimensional Cauchy
problem, in which Cauchy initial data is specified on a three-dimensional
spatial hypersurface, and then evolved forwards in time. The Wolfram model
offers an inherently discrete formulation of the Einstein field equations as an
a priori Cauchy problem, in which Cauchy initial data is specified on a single
spatial hypergraph, and then evolved by means of hypergraph substitution rules,
with the resulting causal network corresponding to the conformal structure of
spacetime. This article introduces a new numerical general relativity code
based upon the conformal and covariant Z4 (CCZ4) formulation with
constraint-violation damping, with the option to reduce to the standard BSSN
formalism if desired, with Cauchy data defined over hypergraphs; the code
incorporates an unstructured generalization of the adaptive mesh refinement
technique proposed by Berger and Colella, in which the topology of the
hypergraph is refined or coarsened based upon local conformal curvature terms.
We validate this code numerically against a variety of standard spacetimes,
including Schwarzschild black holes, Kerr black holes, maximally extended
Schwarzschild black holes, and binary black hole mergers (both rotating and
non-rotating), and explicitly illustrate the relationship between the discrete
hypergraph topology and the continuous Riemannian geometry that is being
approximated. Finally, we compare the results produced by this code to the
results obtained by means of pure Wolfram model evolution (without the
underlying PDE system), using a hypergraph substitution rule that provably
satisfies the Einstein field equations in the continuum limit.
