Entropy-stable summation-by-parts discretization of the Euler equations on general curved elements

Crean, Jared; Hicken, Jason E.; Fernández, David C. Del Rey; Zingg, David W.; Carpenter, Mark H.

doi:10.1016/j.jcp.2017.12.015

Cited by 107 publications

(144 citation statements)

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“…Remarkably, SBP operators can be used to construct high-order spatial discretizations of (9) that also mimic (10). Indeed, an entropy-conservative SBP discretization of (9) is given by [24,26,25,9,45,11,15] du h dt…”

Section: Entropy-conservative Sbp Discretizationmentioning

confidence: 99%

“…Many reasonable choices of A κ are possible that satisfy the assumptions of Theorem 3. In this work I adopt a form for A κ similar to the matrix used to scale the penalty terms in [15]; this choice of A κ is dimensionally consistent. Specifically, I assume A κ is a block diagonal matrix in which the 4 × 4 block corresponding to node i is given by…”

Section: Entropy-stable Continuous Sbp Discretizationmentioning

confidence: 99%

“…The coordinate mapping was then uniquely determined by mapping the Lagrange nodes to physical space. The resulting curvilinear elements were used to define the locations of the SBP nodes (in physical space) and the mapping Jacobian according to [15]. Figure 10(a) shows the p = 2 mesh with N = 4 edges along each boundary, and Figure 10(b) plots the discrete density of the corresponding C-SBP discretization.…”

Section: Accuracy Verification Using the Steady Vortexmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…This was later extended to tensor-product spectral-element methods [9,45] that also possess the summationby-parts property [27]. Subsequently, SBP operators were generalized to simplex elements in [31] and later used to construct entropy-stable discretizations on triangular and tetrahedral grids [11,15].My objective in this paper is to extend the entropy-stable SBP-framework to continuous-Galerkin type discretizations. Previous entropy-stable SBP discretizations have focused on discontinuous-Galerkin (DG)-type methods; even the finite-difference methods in [24], [26], and [25] used numerical flux functions embedded in penalty terms to couple blocks in multiblock grids.…”

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Entropy-Stable, High-Order Summation-by-Parts Discretizations Without Interface Penalties

Hicken

2020

J Sci Comput

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The paper presents high-order accurate, energy-, and entropy-stable discretizations constructed from summation-by-parts (SBP) operators. Notably, the discretizations assemble global SBP operators and use continuous solutions, unlike previous efforts that use discontinuous SBP discretizations. Derivative-based dissipation and local-projection stabilization (LPS) are investigated as options for stabilizing the baseline discretization. These stabilizations are equal up to a multiplicative constant in one dimension, but only LPS remains well conditioned for general, multidimensional SBP operators. Furthermore, LPS is able to take advantage of the additional nodes required by degree 2p diagonal-norms, resulting in an element-local stabilization with a bounded spectral radius. An entropy-stable version of LPS is easily obtained by applying the projection on the entropy variables. Numerical experiments with the linear-advection and Euler equations demonstrate the accuracy, efficiency, and robustness of the stabilized discretizations, and the continuous approach compares favorably with the more common discontinuous SBP methods.Mathematics Subject Classification (2010) 65M06 · 65M60 · 65M70 · 65M12 IntroductionHigh-order discretizations have been put forward as a possible means of improving the efficiency of computational fluid dynamics (CFD) simulations. The arguments in favor of high-order discretizations include both improved accuracy-per-degree-of-freedom as well as better cache usage on current and future architectures. Despite these potential advantages, the use of high-order CFD remains uncommon in industry. Mesh generation of curved This is a pre-print of an article published in The Journal of Scientific Computing. The final authenticated version is available online at: https://doi.bustness: high-order discretizations have inherently less numerical dissipation, which makes them prone to instabilities, particularly for under-resolved flows.Entropy stability offers one promising avenue for constructing robust, high-order CFD methods. This is not a new idea. For example, over thirty years ago, Hughes et al. [36] presented a finite-element discretization of the compressible Navier-Stokes equations that satisfied the second-law of thermodynamics. And in 1999, Barth [2] extended this work to cover Galerkin-least-squares stabilizations and discontinuous Galerkin (DG) schemes. However, these early examples make the assumption that the integrations present in the finite-element semi-linear forms are exact. Exact integration is not possible, in general, for the Euler and Navier-Stokes equations, so these schemes must rely on potentially costly "over-integration" in practice. Even then, the discrete schemes are not provably stable and may fail.In light of the above, there has been growing interest in semi-discrete and fully-discrete high-order schemes that are provably entropy stable. Fisher's thesis [24] represented a seminal contribution in this direction -see also [26] and [25]. He showed that summationby-parts (SBP) fini...

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Section: Entropy-conservative Sbp Discretizationmentioning

confidence: 99%

Section: Entropy-stable Continuous Sbp Discretizationmentioning

confidence: 99%

Section: Accuracy Verification Using the Steady Vortexmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Entropy-Stable, High-Order Summation-by-Parts Discretizations Without Interface Penalties

Hicken

2020

J Sci Comput

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Entropy stable discontinuous Galerkin methods for the shallow water equations with subcell positivity preservation

Wu,

Trask,

Chan

2024

Numerical Methods Partial

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High order schemes are known to be unstable in the presence of shock discontinuities or under‐resolved solution features, and have traditionally required additional filtering, limiting, or artificial viscosity to avoid solution blow up. Entropy stable schemes address this instability by ensuring that physically relevant solutions satisfy a semi‐discrete entropy inequality independently of discretization parameters. However, additional measures must be taken to ensure that solutions satisfy physical constraints such as positivity. In this work, we present a high order entropy stable discontinuous Galerkin (ESDG) method for the nonlinear shallow water equations (SWE) on two‐dimensional (2D) triangular meshes which preserves the positivity of the water heights. The scheme combines a low order positivity preserving method with a high order entropy stable method using convex limiting. This method is entropy stable and well‐balanced for fitted meshes with continuous bathymetry profiles.

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Construction of Modern Robust Nodal Discontinuous Galerkin Spectral Element Methods for the Compressible Navier–Stokes Equations

Winters

Kopriva

Gassner

et al. 2021

CISM International Centre for Mechanical Sciences

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Entropy-stable summation-by-parts discretization of the Euler equations on general curved elements

Cited by 107 publications

References 26 publications

Entropy-Stable, High-Order Summation-by-Parts Discretizations Without Interface Penalties

Entropy-Stable, High-Order Summation-by-Parts Discretizations Without Interface Penalties

Entropy stable discontinuous Galerkin methods for the shallow water equations with subcell positivity preservation

Construction of Modern Robust Nodal Discontinuous Galerkin Spectral Element Methods for the Compressible Navier–Stokes Equations

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