No abstract

is 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. e DGSEM is constructed with a local tensor-product Lagrange-polynomial basis computed from Legendre-Gauss-Loba o (LGL) points. Furthermore, the collocation of interpolation and quadrature nodes is used in the spatial discretization. is 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 ux 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 ux 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 ux function, which satis es the moving mesh entropy condition, is applied in the split form DG framework. is proof does not require any exactness of quadrature in the spatial integrals of the variational form. Nevertheless, entropy conservation is not su cient to tame discontinuities in the numerical solution and thus the entropy conservative moving mesh DGSEM is modi ed by adding numerical dissipation matrices to the entropy conservative uxes. en, 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 speci ed appropriately.Besides the entropy stability, the time discretization of the moving mesh DGSEM will be investigated and it will be proven that the moving mesh DGSEM satis es the free stream preservation property for an arbitrary s-stage Runge-Ku a method. e theoretical properties of the moving mesh DGSEM will be validated by numerical experiments for the compressible Euler equations. Entropy Stable DG Schemes on Moving Meshes where ν is the grid velocity. e chain rule and (2.2) provide J du dt = J ∂u ∂t + ν ∂u ∂ξ ⇔ J ∂u ∂t = J du dt − ν ∂u ∂ξ . (2.3) Hence, by applying the chain rule and rearranging terms the conservation law (2.1) becomes J du dt + ∂f ∂ξ = ν ∂u ∂ξ (2.4) on the reference element. e product rule allows to write the equation (2.4) as J du dt + ∂g ∂ξ = − ∂ν ∂ξ u, (2.5) d dt 1 2

No abstract

No abstract

No abstract

No abstract

No abstract

No abstract