2018/08/24 by Lucas Friedrich, Friedrich, Lucas, Gero Schnücke +9
Earth and Planetary Sciences · Engineering · #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Meteorological Phenomena and Simulations #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1808.08218
openalex publication_date 2018/08/24 · openalex created_date 2022/08/04 · openalex updated_date 2026/07/28
This work examines the development of an entropy conservative (for smooth\nsolutions) or entropy stable (for discontinuous solutions) space-time\ndiscontinuous Galerkin (DG) method for systems of non-linear hyperbolic\nconservation laws. The resulting numerical scheme is fully discrete and\nprovides a bound on the mathematical entropy at any time according to its\ninitial condition and boundary conditions. The crux of the method is that\ndiscrete derivative approximations in space and time are summation-by-parts\n(SBP) operators. This allows the discrete method to mimic results from the\ncontinuous entropy analysis and ensures that the complete numerical scheme\nobeys the second law of thermodynamics. Importantly, the novel method described\nherein does not assume any exactness of quadrature in the variational forms\nthat naturally arise in the context of DG methods. Typically, the development\nof entropy stable schemes is done on the semi-discrete level ignoring the\ntemporal dependence. In this work we demonstrate that creating an entropy\nstable DG method in time is similar to the spatial discrete entropy analysis,\nbut there are important (and subtle) differences. Therefore, we highlight the\ntemporal entropy analysis throughout this work. For the compressible Euler\nequations, the preservation of kinetic energy is of interest besides entropy\nstability. The construction of kinetic energy preserving (KEP) schemes is,\nagain, typically done on the semi-discrete level similar to the construction of\nentropy stable schemes. We present a generalization of the KEP condition from\nJameson to the space-time framework and provide the temporal components for\nboth entropy stability and kinetic energy preservation. The properties of the\nspace-time DG method derived herein is validated through numerical tests for\nthe compressible Euler equations.\n