Entropy-Stable, High-Order Discretizations Using Continuous Summation-By-Parts Operators

Hicken, Jason E.

doi:10.2514/6.2019-3206

Cited by 2 publications

(1 citation statement)

References 24 publications

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“…The staggered-grid variant was discussed in [25,67], and by using this idea, modal DG formulations (evolving polynomials instead of nodal values) were recovered in [7,8]. Continuous SBP operators and the corresponding entropy stable continuous finite element method was developed in [50]. The assumption of conforming simplex meshes can also be greatly relaxed.…”

Section: Introductionmentioning

confidence: 99%

Review of Entropy Stable Discontinuous Galerkin Methods for Systems of Conservation Laws on Unstructured Simplex Meshes

Shu¹

2020

CSIAM-AM

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In this paper, we will build a roadmap for the growing literature of high order quadrature-based entropy stable discontinuous Galerkin (DG) methods, trying to elucidate the motivations and emphasize the contributions. Compared to the classic DG method which is only provably stable for the square entropy, these DG methods can be tailored to satisfy an arbitrary given entropy inequality, and do not require exact integration. The methodology is within the summation-by-parts (SBP) paradigm, such that the discrete operators collocated at the quadrature points should satisfy the SBP property. The construction is relatively easy for quadrature rules with collocated surface nodes. We use the flux differencing technique to ensure entropy balance within elements, and the simultaneous approximation terms (SATs) to produce entropy dissipation on element interfaces. The further extension to general quadrature rules is achieved through careful modifications of SATs.

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Section: Introductionmentioning

confidence: 99%

Review of Entropy Stable Discontinuous Galerkin Methods for Systems of Conservation Laws on Unstructured Simplex Meshes

Shu¹

2020

CSIAM-AM

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Analysis of the SBP-SAT Stabilization for Finite Element Methods Part I: Linear Problems

et al. 2020

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In the hyperbolic community, discontinuous Galerkin (DG) approaches are mainly applied when finite element methods are considered. As the name suggested, the DG framework allows a discontinuity at the element interfaces, which seems for many researchers a favorable property in case of hyperbolic balance laws. On the contrary, continuous Galerkin methods appear to be unsuitable for hyperbolic problems and there exists still the perception that continuous Galerkin methods are notoriously unstable. To remedy this issue, stabilization terms are usually added and various formulations can be found in the literature. However, this perception is not true and the stabilization terms are unnecessary, in general. In this paper, we deal with this problem, but present a different approach. We use the boundary conditions to stabilize the scheme following a procedure that are frequently used in the finite difference community. Here, the main idea is to impose the boundary conditions weakly and specific boundary operators are constructed such that they guarantee stability. This approach has already been used in the discontinuous Galerkin framework, but here we apply it with a continuous Galerkin scheme. No internal dissipation is needed even if unstructured grids are used. Further, we point out that we do not need exact integration, it suffices if the quadrature rule and the norm in the differential operator are the same, such that the summation-by-parts property is fulfilled meaning that a discrete Gauss Theorem is valid. This contradicts the perception in the hyperbolic community that stability issues for pure Galerkin scheme exist. In numerical simulations, we verify our theoretical analysis.

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Entropy-Stable, High-Order Discretizations Using Continuous Summation-By-Parts Operators

Cited by 2 publications

References 24 publications

Review of Entropy Stable Discontinuous Galerkin Methods for Systems of Conservation Laws on Unstructured Simplex Meshes

Review of Entropy Stable Discontinuous Galerkin Methods for Systems of Conservation Laws on Unstructured Simplex Meshes

Analysis of the SBP-SAT Stabilization for Finite Element Methods Part I: Linear Problems

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