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