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Numerical analysis of a canonical shock wave interaction problem in general relativity. (English) Zbl 1333.83022

Summary: We present the analysis of convergence of the locally inertial Godunov method with dynamical time dilation applied to a canonical initial data set which is arguably the simplest initial data that creates a point of shock wave interaction in General Relativity. New applications include the analysis of convergence in the presence of new boundary conditions which enables one to test the validity of the Einstein constraint equations numerically in new Lipschitz continuous space-time metrics. The numerical method, introduced in [the second author, The numerical simulation of general relativistic shock waves by a locally inertial Godunov method featuring dynamic time dilation. Davis, CA: University of California (Diss.) (2010); the authors, “Simulation of general relativistic shock wave interactions by a locally inertial Godunov method featuring dynamical time dilation”, Proc. Royal Soc. A Math. Phys. Eng. Sci. 468, No. 2143, 1865–1883 (2012; doi:10.1098/rspa.2011.0355)], is an algorithm for simulating general relativistic shock-waves in spherically symmetric spacetimes, and the analysis here rigorously establishes claims made in the authors’ PRSA article [loc. cit.].

MSC:

83C10 Equations of motion in general relativity and gravitational theory
76L05 Shock waves and blast waves in fluid mechanics
35L65 Hyperbolic conservation laws
32D20 Removable singularities in several complex variables
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