On discretely entropy conservative and entropy stable discontinuous Galerkin methods

High order methods based on diagonal-norm summation by parts operators can be shown to satisfy a discrete conservation or dissipation of entropy for nonlinear systems of hyperbolic PDEs. These methods can also be interpreted as nodal discontinuous Galerkin methods with diagonal mass matrices. In this work, we describe how use flux differencing, quadrature-based projections, and SBP-like operators to construct discretely entropy conservative schemes for DG methods under more arbitrary choices of volume and surface quadrature rules. The resulting methods are semi-discretely entropy conservative or entropy stable with respect to the volume quadrature rule used. Numerical experiments confirm the stability and high order accuracy of the proposed methods for the compressible Euler equations in one and two dimensions

Configuration polynomials under contact equivalence

Configuration polynomials generalize the classical Kirchhoff polynomial defined by a graph. Their study sheds light on certain polynomials appearing in Feynman integrands. Contact equivalence provides a way to study the associated configuration hypersurface. In the contact equivalence class of any configuration polynomial we identify a polynomial with minimal number of variables; it is a configuration polynomial. This minimal number is bounded by $r+1\choose 2$, where $r$ is the rank of the underlying matroid. We show that the number of equivalence classes is finite exactly up to rank $3$ and list explicit normal forms for these classes.Comment: 19 pages, 1 tabl