  This work is focused on the entropy analysis of a semi-discrete nodal
discontinuous Galerkin spectral element method (DGSEM) on moving meshes for
hyperbolic conservation laws. The DGSEM is constructed with a local
tensor-product Lagrange-polynomial basis computed from Legendre-Gauss-Lobatto
(LGL) points. Furthermore, the collocation of interpolation and quadrature
nodes is used in the spatial discretization. This approach leads to discrete
derivative approximations in space that are summation-by-parts (SBP) operators.
On a static mesh, the SBP property and suitable two-point flux functions, which
satisfy the entropy condition from Tadmor, allow to mimic results from the
continuous entropy analysis on the discrete level. In this paper, Tadmor's
condition is extended to the moving mesh framework. Based on the moving mesh
entropy condition, entropy conservative two-point flux functions for the
homogeneous shallow water equations and the compressible Euler equations are
constructed. Furthermore, it will be proven that the semi-discrete moving mesh
DGSEM is an entropy conservative scheme when a two-point flux function, which
satisfies the moving mesh entropy condition, is applied in the split form DG
framework. This proof does not require any exactness of quadrature in the
spatial integrals of the variational form. Nevertheless, entropy conservation
is not sufficient to tame discontinuities in the numerical solution and thus
the entropy conservative moving mesh DGSEM is modified by adding numerical
dissipation matrices to the entropy conservative fluxes. Then, the method
becomes entropy stable such that the discrete mathematical entropy is bounded
at any time by its initial and boundary data when the boundary conditions are
specified appropriately. The theoretical properties of the moving mesh DGSEM
will be validated by numerical experiments for the compressible Euler
equations.

Numerical Analysis