A discontinuous Galerkin finite element discretization of the Euler equations for compressible and incompressible fluids

Pesch, Lars; Vegt, J.J.W. van der

doi:10.1016/j.jcp.2008.01.046

Cited by 26 publications

(17 citation statements)

References 30 publications

(59 reference statements)

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“…If the corresponding dimensionless variables are used instead of the original ones, the following system is obtained We are interested in the regime when M → 0. This kind of regime may be encountered in low velocity compressible flows (u 0 a 0 ), or in nearly incompressible flows (a 0 → ∞) [45]. Low Mach number flows are also a natural concern when dealing with multiphase compressible flows, in which the sound velocity of the liquid phase is high [30,43,44].…”

Section: Introductionmentioning

confidence: 99%

A low Mach correction able to deal with low Mach acoustics

Bruel

Delmas

Jung

et al. 2019

Journal of Computational Physics

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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

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Section: Introductionmentioning

confidence: 99%

A low Mach correction able to deal with low Mach acoustics

Bruel

Delmas

Jung

et al. 2019

Journal of Computational Physics

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“…More adapted WENO schemes then have been developed [52,23,21] to overcome the problem of convergence toward the steady-state solution. We also mention some interesting developments for the steady-state case based on the Discontinuous Galerkin method [6,7,38], the Spectral Volume technique [10,18], and the Residual Distribution schemes [12,2,22].…”

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confidence: 99%

a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equations

2017

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To cite this version:Stéphane Clain, Raphaël Loubère, Gaspar Machado. a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equations. 2016. a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equationsWe propose a new family of finite volume high-accurate numerical schemes devoted to solve one-dimensional steady-state hyperbolic systems. High-accuracy (up to the sixth-order presently) is achieved thanks to polynomial reconstructions while stability is provided with an a posteriori MOOD method which control the cell polynomial degree for eliminating non-physical oscillations in the vicinity of discontinuities. Such a procedure demands the determination of a chain detector to discriminate between troubled and valid cells, a cascade of polynomial degrees to be successively tested when oscillations are detected, and a parachute scheme corresponding to the last, viscous, and robust scheme of the cascade. Experimented on linear, Burgers', and Euler equations, we demonstrate that the schemes manage to retrieve smooth solutions with optimal order of accuracy but also irregular solutions without spurious oscillations.

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“…In addition, the scheme is accurate to order if the solution is approximated by a polynomial of degree in time. Implicit space-time DG schemes for conservation laws have been studied by several authors (e.g., Bar-Yoseph and Elata 1990, Van der Vegt and Van der Ven 2002, Pesch and Van der Vegt 2008. The common difficulty is that the implicit system is large.…”

Section: Introductionmentioning

confidence: 99%

High-Order Space-Time Methods for Conservation Laws

Huynh

2013

21st AIAA Computational Fluid Dynamics Conference

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Current high-order methods such as discontinuous Galerkin and/or flux reconstruction can provide effective discretization for the spatial derivatives. Together with a time discretization, such methods result in either too small a time step size in the case of an explicit scheme or a very large system in the case of an implicit one. To tackle these problems, two new high-order space-time schemes for conservation laws are introduced: the first is explicit and the second, implicit. The explicit method here, also called the moment scheme, achieves a CFL (Courant-Friedrichs-Lewy) condition of 1 for the case of one-spatial dimension regardless of the degree of the polynomial approximation. (For standard explicit methods, if the spatial approximation is of degree , then the time step sizes are typically proportional to .) Fourier analyses for the one and two-dimensional cases are carried out. The property of super accuracy (or super convergence) is discussed. The implicit method is a simplified but optimal version of the discontinuous Galerkin scheme applied to time. It reduces to a collocation implicit Runge-Kutta (RK) method for ordinary differential equations (ODE) called Radau IIA. The explicit and implicit schemes are closely related since they employ the same intermediate time levels, and the former can serve as a key building block in an iterative procedure for the latter. A limiting technique for the piecewise linear scheme is also discussed. The technique can suppress oscillations near a discontinuity while preserving accuracy near extrema. Preliminary numerical results are shown.

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A discontinuous Galerkin finite element discretization of the Euler equations for compressible and incompressible fluids

Cited by 26 publications

References 30 publications

A low Mach correction able to deal with low Mach acoustics

A low Mach correction able to deal with low Mach acoustics

a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equations

High-Order Space-Time Methods for Conservation Laws

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