Proc. R. Soc. A (2012) 468, 1865–1883 doi:10.1098/rspa.2011.0355 Published online 4 April 2012 Simulation of general relativistic shock wave interactions by a locally inertial Godunov method featuring dynamical time dilation BY ZEKE VOGLER AND BLAKE TEMPLE* Department of Mathematics, University of California, Davis, CA 95616, USA We introduce the locally inertial Godunov method with dynamical time dilation, and use it to give a definitive numerical simulation of a point of shock wave interaction in general relativity starting from a new initial dataset. Prior work of Groah and Temple justifies meeting the Einstein constraint equations for the initial data only at the weak level of Lipshitz continuity in the metric. The forward time simulations, presented here, resolve the secondary 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 solution via collapse associated with an incoming rarefaction wave. A new feature is that space–time is approximated as locally flat in each grid cell so that Riemann problems and the Godunov method can be implemented. Clocks are then dynamically dilated to simulate effects of space–time curvature. Such points of shock wave interaction are more singular than points on single shock surfaces because the coordinate systems that make space–time locally flat on single shock surfaces (Gaussian normal coordinates), break down at points of shock wave interaction. Keywords: locally inertial, Godunov method, general relativity, dynamic time dilation 1. Introduction We summarize the results in the thesis (Vogler 2010) in which Vogler introduces what we term the locally inertial Godunov method with dynamic time dilation, a fractional step method for simulating spherically symmetric shock wave solutions of the Einstein–Euler equations of general relativity (GR) in Standard Schwarzschild Coordinates (SSCs) (Groah & Temple 2004). The underlying issue is that the gravitational metric appears to be singular at shock waves in SSC coordinates—the coordinates in which the Einstein equations take the simplest form (Groah & Temple 2004). The simulations here give a definitive numerical demonstration that the locally inertial Godunov method is nevertheless a viable first-order numerical method for simulating shock waves in SSC. Numerical convergence of the method is demonstrated for a one parameter family of initial data obtained by matching a critically expanding Friedmann–Robertson–Walker (FRW) space–time Lipschitz continuously to the inside of a static Tolmann– Oppenheimer–Volkoff (TOV) solution, creating a point of shock wave interaction *Author for correspondence (temple@math.ucdavis.edu). Received 17 June 2011 Accepted 6 March 2012 This journal is © 2012 The Royal Society 1865 on August 29, 2016 http://rspa.royalsocietypublishing.org/ Downloaded from