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This work is focused on the entropy analysis of a semi-discrete nodaldiscontinuous Galerkin spectral element method (DGSEM) on moving meshes forhyperbolic conservation laws. The DGSEM is constructed with a localtensor-product Lagrange-polynomial basis computed from Legendre-Gauss-Lobatto(LGL) points. Furthermore, the collocation of interpolation and quadraturenodes is used in the spatial discretization. This approach leads to discretederivative approximations in space that are summation-by-parts (SBP) operators.On a static mesh, the SBP property and suitable two-point flux functions, whichsatisfy the entropy condition from Tadmor, allow to mimic results from thecontinuous entropy analysis on the discrete level. In this paper, Tadmor'scondition is extended to the moving mesh framework. Based on the moving meshentropy condition, entropy conservative two-point flux functions for thehomogeneous shallow water equations and the compressible Euler equations areconstructed. Furthermore, it will be proven that the semi-discrete moving meshDGSEM is an entropy conservative scheme when a two-point flux function, whichsatisfies the moving mesh entropy condition, is applied in the split form DGframework. This proof does not require any exactness of quadrature in thespatial integrals of the variational form. Nevertheless, entropy conservationis not sufficient to tame discontinuities in the numerical solution and thusthe entropy conservative moving mesh DGSEM is modified by adding numericaldissipation matrices to the entropy conservative fluxes. Then, the methodbecomes entropy stable such that the discrete mathematical entropy is boundedat any time by its initial and boundary data when the boundary conditions arespecified appropriately. The theoretical properties of the moving mesh DGSEMwill be validated by numerical experiments for the compressible Eulerequations.

Entropy Stable Discontinuous Galerkin Schemes on Moving Meshes with Summation-by-Parts Property for Hyperbolic Conservation Laws