I won’t think much yet about anything other than a Cauchy initial value scheme which proceeds by evolving a 3D space like submanifold of the 4D space time continuum whose equations were given by Einstein.
My grasp of strength of equations is still poor but I am working from the following characterization for empty space which I gleaned from Wald: A space like sub manifold S with its complete 3D metric plus the extrinsic embedding curvature tensor. The latter is a symmetric 3D tensor which provides a quadratic form describing the departure of the submanifold from a tangent plane at a point on S. (I think that the 3D tensor is Γ0ij where coordinate 0 is time and indices j & k range over spatial dimensions.) This suggests that the computational state is a 3D field with two symmetric tensors fields, one of them positive definite.
Combining these images with simplicial ideas I come up with a set of 4D simplexes (zones) which fill in a sandwich between two space like surfaces. The sandwich is just one zone thick in that each zone in the sandwich has vertices in both surfaces. The edge length squared of the zones will probably have both signs. I speculate that at most one of them can be time like lest effects be unable to travel thru the mesh fast enough.
An alternative, possible simpler, initial condition is a zero thickness space-like complex with a value for the stress-energy tensor in each simplex and for each triangle face the total angle about that face in the preceding simplexes. This total captures the extrinsic curvature.