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This work examines the development of an entropy conservative (for smoothsolutions) or entropy stable (for discontinuous solutions) space-timediscontinuous Galerkin (DG) method for systems of non-linear hyperbolicconservation laws. The resulting numerical scheme is fully discrete andprovides a bound on the mathematical entropy at any time according to itsinitial condition and boundary conditions. The crux of the method is thatdiscrete derivative approximations in space and time are summation-by-parts(SBP) operators. This allows the discrete method to mimic results from thecontinuous entropy analysis and ensures that the complete numerical schemeobeys the second law of thermodynamics. Importantly, the novel method describedherein does not assume any exactness of quadrature in the variational formsthat naturally arise in the context of DG methods. Typically, the developmentof entropy stable schemes is done on the semi-discrete level ignoring thetemporal dependence. In this work we demonstrate that creating an entropystable DG method in time is similar to the spatial discrete entropy analysis,but there are important (and subtle) differences. Therefore, we highlight thetemporal entropy analysis throughout this work. For the compressible Eulerequations, the preservation of kinetic energy is of interest besides entropystability. The construction of kinetic energy preserving (KEP) schemes is,again, typically done on the semi-discrete level similar to the construction ofentropy stable schemes. We present a generalization of the KEP condition fromJameson to the space-time framework and provide the temporal components forboth entropy stability and kinetic energy preservation. The properties of thespace-time DG method derived herein is validated through numerical tests forthe compressible Euler equations.

Entropy Stable Space-Time Discontinuous Galerkin Schemes with Summation-by-Parts Property for Hyperbolic Conservation Laws