Comparison of Some Entropy Conservative Numerical Fluxes for the Euler Equations

Ranocha, Hendrik

doi:10.1007/s10915-017-0618-1

Cited by 76 publications

(67 citation statements)

References 27 publications

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“…from which we can see that W V q L q diag( n i )V f P q is symmetric. Combining (36) and (37), we have that the top left block of…”

Section: Quadrature-based Projection Matrices and Lifting Matricesmentioning

confidence: 99%

“…In order to generalize beyond diagonal norm SBP-DG methods, we will consider DG discretizations using over-integrated quadrature rules with more points than the dimension of the approximation space, which are commonly used for non-tensor product elements in two and three dimensions [35]. These quadrature rules induce DG schemes which are related to dense norm and generalized SBP operators [36,28,37], for which discretely entropy stable schemes for the compressible Euler equations have not yet been constructed. We present proofs of discrete entropy stability using both a matrix formulation involving a "decoupled" SBP-like operator and continuous formulations involving projection and lifting operators.…”

Section: Introductionmentioning

confidence: 99%

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On discretely entropy conservative and entropy stable discontinuous Galerkin methods

Chan

2018

Journal of Computational Physics

142

217

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High order methods based on diagonal-norm summation by parts operators can be shown to satisfy a discrete conservation or dissipation of entropy for nonlinear systems of hyperbolic PDEs [1,2]. These methods can also be interpreted as nodal discontinuous Galerkin methods with diagonal mass matrices [3,4,5,6]. In this work, we describe how use flux differencing, quadrature-based projections, and SBP-like operators to construct discretely entropy conservative schemes for DG methods under more arbitrary choices of volume and surface quadrature rules. The resulting methods are semi-discretely entropy conservative or entropy stable with respect to the volume quadrature rule used. Numerical experiments confirm the stability and high order accuracy of the proposed methods for the compressible Euler equations in one and two dimensions.The focus of this work is the construction of high order polynomial DG methods which satisfy a discrete analogue of the conservation of entropy (5) and the dissipation of entropy (7). Discrete differential operators and quadrature-based matrices Mathematical assumptions and notationsWe begin with a d-dimensional reference element D with boundary ∂ D. We denote the ith component of the outward normal vector on the boundary of the reference element ∂ D as n i . For this work, we assume

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“…from which we can see that W V q L q diag( n i )V f P q is symmetric. Combining (36) and (37), we have that the top left block of…”

Section: Quadrature-based Projection Matrices and Lifting Matricesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On discretely entropy conservative and entropy stable discontinuous Galerkin methods

Chan

2018

Journal of Computational Physics

142

217

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“…There exist several choices for entropy conservative fluxes [14,46,47]. We utilize the the entropy conservative numerical fluxes given by Chandrashekar in [14]…”

Section: Numerical Experimentsmentioning

confidence: 99%

Skew-Symmetric Entropy Stable Modal Discontinuous Galerkin Formulations

Chan

2019

J Sci Comput

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High order entropy stable discontinuous Galerkin (DG) methods for nonlinear conservation laws satisfy an inherent discrete entropy inequality. The construction of such schemes has relied on the use of carefully chosen nodal points [1][2][3][4][5] or volume and surface quadrature rules [6,7] to produce operators which satisfy a summation-by-parts (SBP) property. In this work, we show how to construct "modal" DG formulations which are entropy stable for volume and surface quadratures under which the SBP property in [6] does not hold. These formulations rely on an alternative skew-symmetric construction of operators which automatically satisfy the SBP property. Entropy stability then follows for choices of volume and surface quadrature which satisfy sufficient accuracy conditions. The accuracy of these new SBP operators depends on a separate set of conditions on quadrature accuracy, with design order accuracy recovered under the usual assumptions of degree 2N −1 volume quadratures and degree 2N surface quadratures. We conclude with numerical experiments verifying the accuracy and stability of the proposed formulations, and discuss an application of these formulations for entropy stable DG schemes on mixed quadrilateral-triangle meshes. IntroductionHigh order methods for the simulation of time-dependent compressible flow have the potential to achieve higher levels of accuracy at lower costs compared to current low order schemes [8]. In addition to superior accuracy, the low numerical dispersion and dissipation of high order methods [9] enables the accurate propagation of waves over long distances and time scales. The same properties also make high order methods attractive for unsteady phenomena such as vorticular and turbulent flows, which are sensitive to numerical dissipation [8,10].

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“…Entropy-conservative and entropy-stable numerical fluxes. There has been much recent interest in the development of entropy-conservative and entropy-stable numerical flux functions for the Euler equations [17,31,5,33]. In this work, for the volume fluxes, we will use the two-point entropyconservative flux of Chandrashekar [5].…”

Section: 21mentioning

confidence: 99%

Analysis and Entropy Stability of the Line-Based Discontinuous Galerkin Method

Pazner

Persson

2019

J Sci Comput

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We develop a discretely entropy-stable line-based discontinuous Galerkin method for hyperbolic conservation laws based on a flux differencing technique. By using standard entropystable and entropy-conservative numerical flux functions, this method guarantees that the discrete integral of the entropy is non-increasing. This nonlinear entropy stability property is important for the robustness of the method, in particular when applied to problems with discontinuous solutions or when the mesh is under-resolved. This line-based method is significantly less computationally expensive than a standard DG method. Numerical results are shown demonstrating the effectiveness of the method on a variety of test cases, including Burgers' equation and the Euler equations, in one, two, and three spatial dimensions. arXiv:1809.09815v2 [math.NA]

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Comparison of Some Entropy Conservative Numerical Fluxes for the Euler Equations

Cited by 76 publications

References 27 publications

On discretely entropy conservative and entropy stable discontinuous Galerkin methods

On discretely entropy conservative and entropy stable discontinuous Galerkin methods

Skew-Symmetric Entropy Stable Modal Discontinuous Galerkin Formulations

Analysis and Entropy Stability of the Line-Based Discontinuous Galerkin Method

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