Spectral Analysis of Continuous FEM for Hyperbolic PDEs: Influence of Approximation, Stabilization, and Time-Stepping

Michel, Sixtine; Torlo, Davide; Ricchiuto, Mario; Abgrall, Rémi

doi:10.1007/s10915-021-01632-7

Cited by 9 publications

(25 citation statements)

References 42 publications

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“…The methodology we explain in the following, will be applied to all the combination of schemes we presented above (in time: RK, SSPRK and DeC, discretisation in space: Basic, Cubature and Bernstein, stabilization techniques: CIP, OSS and SUPG), in order to find the best coefficients (CFL, δ), as in [42].…”

Section: Methodsmentioning

confidence: 99%

“…What we aim to do is an optimization process also on the stabilization parameter and the CFL number. With the notation of [42], we will set for the different stabilizations…”

Section: Methodsmentioning

confidence: 99%

“…Unfortunately, as already mentioned above, for some schemes the stability area is not so well defined for several reasons. One of these reasons is the "shape" of the stability area as for one-dimensional problems, see [42]. Other issues that affect this analysis are the numerical precision, see Section 3.6, and the mesh configuration, see Section 3.7.…”

Section: Methodsmentioning

confidence: 99%

“…in this example, we consider the SSPRK discretization (42). From Equation (43) we can write the amplification coefficient Γ(λ), i.e.,…”

Section: Comparison With a Space-time Split Stability Analysismentioning

confidence: 99%

See 3 more Smart Citations

Spectral analysis of high order continuous FEM for hyperbolic PDEs on triangular meshes: influence of approximation, stabilization, and time-stepping

Michel¹,

Torlo²,

Ricchiuto³

et al. 2022

Preprint

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In this work we study various continuous finite element discretization for two dimensional hyperbolic partial differential equations, varying the polynomial space (Lagrangian on equispaced, Lagrangian on quadrature points (Cubature) and Bernstein), the stabilization techniques (streamline-upwind Petrov-Galerkin, continuous interior penalty, orthogonal subscale stabilization) and the time discretization (Runge-Kutta (RK), strong stability preserving RK and deferred correction). This is an extension of the one dimensional study by Michel S. et al J. Sci. Comput. (2021), whose results do not hold in multi-dimensional frameworks. The study ranks these schemes based on efficiency (most of them are mass-matrix free), stability and dispersion error, providing the best CFL and stabilization coefficients. The challenges in two-dimensions are related to the Fourier analysis. Here, we perform it on two types of periodic triangular meshes varying the angle of the advection, and we combine all the results for a general stability analysis. Furthermore, we introduce additional high order viscosity to stabilize the discontinuities, in order to show how to use these methods for tests of practical interest. All the theoretical results are thoroughly validated numerically both on linear and non-linear problems, and error-CPU time curves are provided. Our final conclusions suggest that Cubature elements combined with SSPRK and OSS stabilization is the most promising combination.

show abstract

Section: Methodsmentioning

confidence: 99%

“…What we aim to do is an optimization process also on the stabilization parameter and the CFL number. With the notation of [42], we will set for the different stabilizations…”

Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

“…in this example, we consider the SSPRK discretization (42). From Equation (43) we can write the amplification coefficient Γ(λ), i.e.,…”

Section: Comparison With a Space-time Split Stability Analysismentioning

confidence: 99%

See 2 more Smart Citations

Spectral analysis of high order continuous FEM for hyperbolic PDEs on triangular meshes: influence of approximation, stabilization, and time-stepping

Michel¹,

Torlo²,

Ricchiuto³

et al. 2022

Preprint

Self Cite

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show abstract

“…are equivalent, because for σ ∈ K, S σ is contained in the set of DOFs in K, and those of the neighbouring elements to K. We recall that λ depends on the maximum of U h over the mesh which is bounded. We point out that (38) together with (39) yields the BV estimate, namely…”

Section: By Introducing a Correction Of The Form αmentioning

confidence: 99%

On the convergence of residual distribution schemes for the compressible Euler equations via dissipative weak solutions

Abgrall¹,

Lukácova-Medvid'ová²,

Öffner³

2022

Preprint

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In this work, we prove the convergence of residual distribution schemes to dissipative weak solutions of the Euler equations. We need to guarantee that the residual distribution schemes are fulfilling the underlying structure preserving properties such as positivity of density and internal energy. Consequently, the residual distribution schemes lead to a consistent and stable approximation of the Euler equations. Our result can be seen as a generalization of the Lax-Richtmyer equivalence theorem to nonlinear problems that consistency plus stability is equivalent to convergence.

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Spectral Analysis of High Order Continuous FEM for Hyperbolic PDEs on Triangular Meshes: Influence of Approximation, Stabilization, and Time-Stepping

et al. 2023

Self Cite

View full text Add to dashboard Cite

In this work we study various continuous finite element discretization for two dimensional hyperbolic partial differential equations, varying the polynomial space (Lagrangian on equispaced, Lagrangian on quadrature points (Cubature) and Bernstein), the stabilization techniques (streamline-upwind Petrov–Galerkin, continuous interior penalty, orthogonal subscale stabilization) and the time discretization (Runge–Kutta (RK), strong stability preserving RK and deferred correction). This is an extension of the one dimensional study by Michel et al. (J Sci Comput 89(2):31, 2021. https://doi.org/10.1007/s10915-021-01632-7), whose results do not hold in multi-dimensional frameworks. The study ranks these schemes based on efficiency (most of them are mass-matrix free), stability and dispersion error, providing the best CFL and stabilization coefficients. The challenges in two-dimensions are related to the Fourier analysis. Here, we perform it on two types of periodic triangular meshes varying the angle of the advection, and we combine all the results for a general stability analysis. Furthermore, we introduce additional high order viscosity to stabilize the discontinuities, in order to show how to use these methods for tests of practical interest. All the theoretical results are thoroughly validated numerically both on linear and non-linear problems, and error-CPU time curves are provided. Our final conclusions suggest that Cubature elements combined with SSPRK and OSS stabilization is the most promising combination.

show abstract

Spectral Analysis of Continuous FEM for Hyperbolic PDEs: Influence of Approximation, Stabilization, and Time-Stepping

Cited by 9 publications

References 42 publications

Spectral analysis of high order continuous FEM for hyperbolic PDEs on triangular meshes: influence of approximation, stabilization, and time-stepping

Spectral analysis of high order continuous FEM for hyperbolic PDEs on triangular meshes: influence of approximation, stabilization, and time-stepping

On the convergence of residual distribution schemes for the compressible Euler equations via dissipative weak solutions

Spectral Analysis of High Order Continuous FEM for Hyperbolic PDEs on Triangular Meshes: Influence of Approximation, Stabilization, and Time-Stepping

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