Entropy Stable Space-Time Discontinuous Galerkin Schemes with Summation-by-Parts Property for Hyperbolic Conservation Laws

Friedrich, Lucas; Schnücke, Gero; Winters, Andrew R.; Fernández, David C. Del Rey; Gassner, Gregor J.; Carpenter, Mark H.

doi:10.48550/arxiv.1808.08218

Cited by 4 publications

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“…in energy stable schemes for nonlinear equations [37]. This is also related to space-time discontinuous Galerkin schemes, where entropy stability can be obtained [15].…”

Section: Summary and Discussionmentioning

confidence: 96%

On strong stability of explicit Runge–Kutta methods for nonlinear semibounded operators

Ranocha

2020

IMA Journal of Numerical Analysis

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Many important initial value problems have the property that energy is nonincreasing in time. Energy stable methods, also referred to as strongly stable methods, guarantee the same property discretely. We investigate requirements for conditional energy stability of explicit Runge-Kutta methods for nonlinear or non-autonomous problems. We provide both necessary and sufficient conditions for energy stability over these classes of problems. Examples of conditionally energy stable schemes are constructed and an example is given in which unconditional energy stability is obtained with an explicit scheme. IntroductionEver since the construction of numerical methods for ordinary and (time-dependent) partial differential equations (ODEs and PDEs, respectively), their stability has been an important and active topic of research. Monotonicity, meaning that the norm of the solution is bounded by its initial value, is a particularly exacting stability property. For equations, such as parabolic PDEs, that contain a significant amount of dissipation, any reasonable numerical method will typically preserve monotonicity under an appropriate time step restriction. In contrast, for non-dissipative problems such as hyperbolic PDEs (and their slightly dissipative semidiscretizations), common time discretizations may not preserve monotonicity under any finite step size.The energy method is an effective tool to get stability estimates, e.g. for hyperbolic PDEs [12,19]. Using summation by parts operators [9,34], these can be transferred efficiently to the semidiscrete level for many different kinds of schemes [11,20,21,28]. However, applying the same approach in time yields implicit methods [2,10,22,25]. Classical nonlinearly stable methods, such as algebraically stable Runge-Kutta methods, are also implicit. For hyperbolic problems, such implicit methods are usually less efficient than explicit ones. It is possible to obtain conditional energy stability with explicit methods by using modifications that go outside the class of Runge-Kutta methods; e.g. projection methods [5,6,13] and relaxation Runge-Kutta schemes [18,29].Nevertheless, it is interesting to know what can be achieved within the class of explicit Runge-Kutta methods without modifications. In this setting, results have been obtained for problems that include a certain amount of dissipation [8,16]. Recently this topic has again attracted the interest of researchers and several results (using the term strong stability) for linear, time-independent operators have been discovered [27,32,33]. Nonlinear problems have been investigated in [24], where many non-existence results for energy stable and strong stability 1 arXiv:1909.13215v1 [math.NA]

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“…in energy stable schemes for nonlinear equations [37]. This is also related to space-time discontinuous Galerkin schemes, where entropy stability can be obtained [15].…”

Section: Summary and Discussionmentioning

confidence: 96%

On strong stability of explicit Runge–Kutta methods for nonlinear semibounded operators

Ranocha

2020

IMA Journal of Numerical Analysis

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“…State functions u # with the property (2.23) have been constructed by Friedrich et al [23] for the shallow water, compressible Euler and ideal magnetohydrodynamic (MHD) equations and are needed for the construction of fully discrete entropy stable space-time DGSEM methods.…”

Section: Discrete Entropy Conservation For Moving Mesh Fv Methodsmentioning

confidence: 99%

Entropy Stable Discontinuous Galerkin Schemes on Moving Meshes for Hyperbolic Conservation Laws

et al. 2020

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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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“…In practice, we observe that entropy is dissipated for all problems considered, despite the fact that the proof does not hold at the fully discrete level. A fully discrete entropy inequality can be guaranteed using implicit or space-time discretizations [26,27].…”

Section: Decoupled Sbp Operatorsmentioning

confidence: 99%

Efficient Entropy Stable Gauss Collocation Methods

Chan¹,

Fernández²,

Carpenter³

2019

SIAM J. Sci. Comput.

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105

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The construction of high order entropy stable collocation schemes on quadrilateral and hexahedral elements has relied on the use of Gauss-Legendre-Lobatto collocation points [1,2,3] and their equivalence with summation-by-parts (SBP) finite difference operators [4]. In this work, we show how to efficiently generalize the construction of semi-discretely entropy stable schemes on tensor product elements to Gauss points and generalized SBP operators. Numerical experiments suggest that the use of Gauss points significantly improves accuracy on curved meshes.

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Entropy Stable Space-Time Discontinuous Galerkin Schemes with Summation-by-Parts Property for Hyperbolic Conservation Laws

Cited by 4 publications

References 0 publications

On strong stability of explicit Runge–Kutta methods for nonlinear semibounded operators

On strong stability of explicit Runge–Kutta methods for nonlinear semibounded operators

Entropy Stable Discontinuous Galerkin Schemes on Moving Meshes for Hyperbolic Conservation Laws

Efficient Entropy Stable Gauss Collocation Methods

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