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

Winters, Andrew R.; Kopriva, David A.; Gassner, Gregor J.; Hindenlang, Florian

doi:10.1007/978-3-030-60610-7_3

Cited by 14 publications

(34 citation statements)

References 97 publications

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“…Equation (21) shows that the energy is dissipated by the interface only if a R > a L . Otherwise the interface generates energy, as illustrated in Fig.…”

Section: Energy Dynamics Of the Scalar Problem In One Space Dimensionmentioning

confidence: 99%

“…Equivalence of the norms means that the L 2 norm is actually bounded in terms of the initial data, even though the energy method does not show it directly through (21). From (23) and (31),…”

Section: Remark 24mentioning

confidence: 99%

“…represent the volume and surface quadratures, see [11]. The right hand side of (70) is written in terms of the normal covariant fluxes and is equivalent to that written in terms of the contravariant ones [21]. The resulting high-order semi-discretization is integrated with a proper high-order accurate explicit Runge-Kutta time integrator, which is stable under the typical CFL-type time step restriction.…”

Section: The Discontinuous Galerkin Spectral Element Discretizationmentioning

confidence: 99%

“…For a detailed derivation of the discrete stability estimate, which parallels the continuous analysis, we refer to [4,21]. Here, we will only sketch some important intermediate steps.…”

Section: Discrete Stability Estimatementioning

confidence: 99%

“…Using the fact that the symmetrizer matrix S commutes with the metric block matrix M (see e.g. [4,21]) we see that the volume terms cancel, leaving only surface terms,…”

Section: Discrete Stability Estimatementioning

confidence: 99%

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Stability of Discontinuous Galerkin Spectral Element Schemes for Wave Propagation when the Coefficient Matrices have Jumps

2021

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We use the behavior of the $$L_{2}$$ L 2 norm of the solutions of linear hyperbolic equations with discontinuous coefficient matrices as a surrogate to infer stability of discontinuous Galerkin spectral element methods (DGSEM). Although the $$L_{2}$$ L 2 norm is not bounded in terms of the initial data for homogeneous and dissipative boundary conditions for such systems, the $$L_{2}$$ L 2 norm is easier to work with than a norm that discounts growth due to the discontinuities. We show that the DGSEM with an upwind numerical flux that satisfies the Rankine–Hugoniot (or conservation) condition has the same energy bound as the partial differential equation does in the $$L_{2}$$ L 2 norm, plus an added dissipation that depends on how much the approximate solution fails to satisfy the Rankine–Hugoniot jump.

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“…Equation (21) shows that the energy is dissipated by the interface only if a R > a L . Otherwise the interface generates energy, as illustrated in Fig.…”

Section: Energy Dynamics Of the Scalar Problem In One Space Dimensionmentioning

confidence: 99%

Section: Remark 24mentioning

confidence: 99%

Section: The Discontinuous Galerkin Spectral Element Discretizationmentioning

confidence: 99%

“…For a detailed derivation of the discrete stability estimate, which parallels the continuous analysis, we refer to [4,21]. Here, we will only sketch some important intermediate steps.…”

Section: Discrete Stability Estimatementioning

confidence: 99%

“…Using the fact that the symmetrizer matrix S commutes with the metric block matrix M (see e.g. [4,21]) we see that the volume terms cancel, leaving only surface terms,…”

Section: Discrete Stability Estimatementioning

confidence: 99%

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Stability of Discontinuous Galerkin Spectral Element Schemes for Wave Propagation when the Coefficient Matrices have Jumps

2021

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The Flux-Differencing Discontinuous Galerkin Method Applied to an Idealized Fully Compressible Nonhydrostatic Dry Atmosphere

Souza

Bischoff

et al. 2022

Preprint

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An Entropy Stable Discontinuous Galerkin Method for the Two-Layer Shallow Water Equations on Curvilinear Meshes

Ersing,

Winters

2024

J Sci Comput

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We present an entropy stable nodal discontinuous Galerkin spectral element method (DGSEM) for the two-layer shallow water equations on two dimensional curvilinear meshes. We mimic the continuous entropy analysis on the semi-discrete level with the DGSEM constructed on Legendre–Gauss–Lobatto (LGL) nodes. The use of LGL nodes endows the collocated nodal DGSEM with the summation-by-parts property that is key in the discrete analysis. The approximation exploits an equivalent flux differencing formulation for the volume contributions, which generate an entropy conservative split-form of the governing equations. A specific combination of a numerical surface flux and discretization of the nonconservative terms is then applied to obtain a high-order path-conservative scheme that is entropy conservative. Furthermore, we find that this combination yields an analogous discretization for the pressure and nonconservative terms such that the numerical method is well-balanced for discontinuous bathymetry on curvilinear domains. Dissipation is added at the interfaces to create an entropy stable approximation that satisfies the second law of thermodynamics in the discrete case, while maintaining the well-balanced property. We conclude with verification of the theoretical findings through numerical tests and demonstrate results about convergence, entropy stability and well-balancedness of the scheme.

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

Cited by 14 publications

References 97 publications

Stability of Discontinuous Galerkin Spectral Element Schemes for Wave Propagation when the Coefficient Matrices have Jumps

Stability of Discontinuous Galerkin Spectral Element Schemes for Wave Propagation when the Coefficient Matrices have Jumps

The Flux-Differencing Discontinuous Galerkin Method Applied to an Idealized Fully Compressible Nonhydrostatic Dry Atmosphere

An Entropy Stable Discontinuous Galerkin Method for the Two-Layer Shallow Water Equations on Curvilinear Meshes

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