A Proof of Convergence for Numerical Approximations Generated by the Locally Inertial Godunov Method in General Relativity

In this paper we fill in the details in the proof of convergence stated in Section 7 of [14], for the locally inertial Godunov method with dynamic time dilation, a numerical method for computing shock wave solutions of the Einstein equations in Standard Schwarzschild Coordinates (SSC). We refer the reader to [14] for an introduction and for the notation assumed at the start here. The main conclusion of the theorem stated and proved below is that a sequence of approximate solutions (u∆x,A∆x) → (u,A) of the locally inertial Godunov scheme that converge boundedly without oscillation to a limit function (u,A), must be a weak solution of the Einstein Euler equations in SSC, c.f. [14, 2],