We introduce what we call a locally inertial Godunov method with dynamical time dilation, and use it to simulate a new one parameter family of general relativistic shock wave solutions of the Einstein equations for a perfect fluid. The forward time solutions resolve the secondary reflected wave (an incoming shock wave) in the Smoller-Temple shock wave model for an explosion into a static singular isothermal sphere. The backward time solutions indicate black hole formation from a smooth underlying solution via collapse associated with an incoming rarefaction wave. As far as we know, this is the first numerical simulation of a fluid dynamical shock wave in general relativity.

We provide details for the proof of convergence of the locally inertial Godunov method with dynamical time dilation.

We introduce a new asymptotic ansatz for spherical perturbations of the Standard Model of Cosmology (SM) which applies during the $p=0$ epoch, and prove that these perturbations trigger instabilities in the SM on the scale of the supernova data. These instabilities create a large, central region of uniform under-density which expands faster than the SM, and this central region of accelerated uniform expansion introduces into the SM {\it precisely} the same range of corrections to redshift vs luminosity as are produced by the cosmological constant in the theory of Dark Energy. A universal behavior is exhibited because all sufficiently small perturbations evolve to a single stable rest point. Moreover, we prove that these perturbations are consistent with, and the instability is triggered by, the one parameter family of self-similar waves which the authors previously proposed as possible time-asymptotic wave patterns for perturbations of the SM at the end of the radiation epoch. Using numerical simulations, we calculate the unique wave in the family that accounts for the same values of the Hubble constant and quadratic correction to redshift vs luminosity as in a universe with seventy percent Dark Energy, $\Omega_{\Lambda}\approx.7$. A numerical simulation of the third order correction associated with that unique wave establishes a testable prediction that distinguishes this theory from the theory of Dark Energy. This explanation for the anomalous acceleration, based on instabilities in the SM together with simple wave perturbations from the radiation epoch that trigger them, provides perhaps the simplest mathematical explanation for the anomalous acceleration of the galaxies that does not invoke Dark Energy.