This work examines the development of an entropy conservative (for smooth solutions) or entropy stable (for discontinuous solutions) space-time discontinuous Galerkin (DG) method for systems of non-linear hyperbolic conservation laws. The resulting numerical scheme is fully discrete and provides a bound on the mathematical entropy at any time according to its initial condition and boundary conditions. The crux of the method is that discrete derivative approximations in space and time are summation-by-parts (SBP) operators. This allows the discrete method to mimic results from the continuous entropy analysis and ensures that the complete numerical scheme obeys the second law of thermodynamics. Importantly, the novel method described herein does not assume any exactness of quadrature in the variational forms that naturally arise in the context of DG methods. Typically, the development of entropy stable schemes is done on the semi-discrete level ignoring the temporal dependence. In this work we demonstrate that creating an entropy stable DG method in time is similar to the spatial discrete entropy analysis, but there are important (and subtle) differences. Therefore, we highlight the temporal entropy analysis throughout this work. For the compressible Euler equations, the preservation of kinetic energy is of interest besides entropy stability. The construction of kinetic energy preserving (KEP) schemes is, again, typically done on the semi-discrete level similar to the construction of entropy stable schemes. We present a generalization of the KEP condition from Jameson to the space-time framework and provide the temporal components for both entropy stability and kinetic energy preservation. The properties of the space-time DG method derived herein is validated through numerical tests for the compressible Euler equations. Additionally, we provide, in appendices, how to construct the temporal entropy stable components for the shallow water or ideal magnetohydrodynamic (MHD) equations.
In this paper, we develop and analyze an arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) method with a time-dependent approximation space for one dimensional conservation laws, which satisfies the geometric conservation law. For the semi-discrete ALE-DG method, when applied to nonlinear scalar conservation laws, a cell entropy inequality, L 2 stability and error estimates are proven. More precisely, we prove the sub-optimal (k + 1 2 ) convergence for monotone fluxes and optimal (k + 1) convergence for an upwind flux when a piecewise P k polynomial approximation space is used. For the fully-discrete ALE-DG method, the geometric conservation law and the local maximum principle are proven. Moreover, we state conditions for slope limiters, which ensure total variation stability of the method. Numerical examples show the capability of the method.
In Klingenberg, Schnücke and Xia (Math. Comp. 86 (2017), 1203-1232) an arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) method to solve conservation laws has been developed and analyzed. In this paper, the ALE-DG method will be extended to several dimensions. The method will be designed for simplex meshes. This will ensure that the method satisfies the geometric conservation law, if the accuracy of the time integrator is not less than the value of the spatial dimension. For the semi-discrete method the L 2 -stability will be proven. Furthermore, an error estimate which provides the suboptimal (k + 1 2 ) convergence with respect to the L ∞ 0, T ; L 2 (Ω) -norm will be presented, when an arbitrary monotone flux is used and for each cell the approximating functions are given by polynomials of degree k. The two dimensional fully-discrete explicit method will be combined with the bound preserving limiter developed by Zhang, Xia and Shu in (J. Sci. Comput. 50 (2012), 29-62). This limiter does not affect the high order accuracy of a numerical method. Then, for the ALE-DG method revised by the limiter the validity of a discrete maximum principle will be proven. The numerical stability, robustness and accuracy of the method will be shown by a variety of two dimensional computational experiments on moving triangularThe numerical flux function needs to satisfy certain properties. These properties are discussed in the Section 2.4.Finally, on the reference cell, the semi-discrete ALE-DG method appears as the following problem:Problem 1 (The semi-discrete ALE-DG method on the reference cell). Find a function(4.5)Proof. Since f (c) contains merely constant coefficients andthe integration by parts formula providesThus, we obtain the identity (4.5) by (4.6) and (4.7).Next, we assume that u * h = c solves the semi-discrete ALE-DG method Problem 1. Then, we obtain by (4.4) and (4.5)(4.8) 17The equation (4.8) and the ODE (2.12) are equivalent, since c is an arbitrary constant,is an arbitrary test function and the quantitiesare merely time-dependent. We note that the time evolution of the metric terms J K(t) needs to be respected in the time discretization of the semi-discrete ALE-DG method Problem 1. Therefore, we discretize the ODE (2.12) and (4.4) simultaneously by the same TVD-RK method. The stage solutions of the TVD-RK discretization for (2.12) will be used to update the metric terms in the TVD-RK discretization for (4.4).The fully-discrete ALE-DG method: First, the ODE (2.12) is discretized by a s-stage TVD-RK method:where K n+γ j := K (t n + γ j t) and ω n+γ j := ω (t n + γ j t). The stage solutions {J K n,i } s i=0 are used to update the metric terms in the TVD-RK discretization of (4.4). The Runge-Kutta method needs to solve the ODE (2.12) exact such thatwhere T (t n+1 ) is the regular mesh of simplices which has been used in the Section 2.1 to construct the time-dependent cells (2.3). We note that in a d-dimensional space a TVD-RK method with order greater than or equal to d is necessary to compute the metric ...
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
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