A Kinetic Energy Preserving DG Scheme Based on Gauss-Legendre Points

In the context of numerical methods for conservation laws, not only the preservation of the primary conserved quantities can be of interest, but also the balance of secondary ones such kinetic energy in case of the Euler equations of gas dynamics. In this work, we construct a kinetic energy preserving discontinuous Galerkin method on Gauss-Legendre nodes based on the framework of summation-by-parts operators. For a Gauss-Legendre point distribution, boundary terms require special attention. In fact, stability problems will be demonstrated for a combination of skew-symmetric and boundary terms that disagrees with exclusively interior nodal sets. We will theoretically investigate the required form of the corresponding boundary correction terms in the skew-symmetric formulation leading to a conservative and consistent scheme. In numerical experiments, we study the order of convergence for smooth solutions, the kinetic energy balance and the behaviour of different variants of the scheme applied to an acoustic pressure wave and a viscous shock tube. Using Gauss-Legendre nodes results in a more accurate approximation in our numerical experiments for viscous compressible flow. Moreover, for two-dimensional decaying homogeneous turbulence, kinetic energy preservation yields a better representation of the energy spectrum.