Relaxation of Energy and Approximate Riemann Solvers for General Pressure Laws in Fluid Dynamics

Coquel, Fré́dé́ric; Perthame, Benôıt

doi:10.1137/s0036142997318528

Cited by 158 publications

(172 citation statements)

References 15 publications

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“…On the same test case as above, we obtained the results of Figures 5,6,7,8,9. We observe that the pressure remains positive and that a vapor bubble appears in the center of the computational domain, but this is only a qualitative validation. We observe small perturbations of the pressure on Figure 7.…”

Section: Numerical Results For the Transition Casementioning

confidence: 88%

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Finite volume simulation of cavitating flows

Barberon¹,

Helluy²

2005

Computers & Fluids

110

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We propose a numerical method adapted to the modelling of phase transitions in compressible fluid flows. Pressure laws taking into account phase transitions are complex and lead to difficulties such as the non-uniqueness of entropy solutions. In order to avoid these difficulties, we propose a projection finite volume scheme. This scheme is based on a Riemann solver with a simpler pressure law and an entropy maximization procedure that enables us to recover the original complex pressure law. Several numerical experiments are presented that validate this approach.

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Section: Numerical Results For the Transition Casementioning

confidence: 88%

“…The idea of replacing a complex pressure law by a simpler one together with a relaxation procedure was first proposed by Coquel and Perthame in [9]. Relaxation source terms are extensively used in the seven-equation model of Abgrall and Saurel in [26].…”

Section: Introductionmentioning

confidence: 99%

Finite volume simulation of cavitating flows

Barberon¹,

Helluy²

2005

Computers & Fluids

110

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“…Our approach was in particular suggested by the recent relaxation solver of Berthon and Marche [21], but the main ideas follow primarily the work of Jin and Xin [29] and Suliciu [18,19]. Other related works are for instance [30,31,23]. We refer in particular to the monograph [20] and the bibliography therein.…”

Section: Relaxation Methods For the Single-phase Shallow Flow Modelmentioning

confidence: 99%

“…Both our relaxation model and the one in [21] are based on the pioneering idea of Jin and Xin [29] of approximating the original model equations via a new system that is easier to solve (see also e.g. [30,31,23]). The particular feature that we have borrowed from Berthon and Marche [21] is the formulation of a relaxation system with linear degeneracy in all the characteristic fields, a property obtained by a special decoupling of the linear equations governing the relaxation variables from the remaining non-linear equations of the relaxation model.…”

Section: Introductionmentioning

confidence: 99%

A Riemann solver for single-phase and two-phase shallow flow models based on relaxation. Relations with Roe and VFRoe solvers

Pelanti

Bouchut

Mangeney

2011

Journal of Computational Physics

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a b s t r a c tWe present a Riemann solver derived by a relaxation technique for classical single-phase shallow flow equations and for a two-phase shallow flow model describing a mixture of solid granular material and fluid. Our primary interest is the numerical approximation of this two-phase solid/fluid model, whose complexity poses numerical difficulties that cannot be efficiently addressed by existing solvers. In particular, we are concerned with ensuring a robust treatment of dry bed states. The relaxation system used by the proposed solver is formulated by introducing auxiliary variables that replace the momenta in the spatial gradients of the original model systems. The resulting relaxation solver is related to Roe solver in that its Riemann solution for the flow height and relaxation variables is formally computed as Roe's Riemann solution. The relaxation solver has the advantage of a certain degree of freedom in the specification of the wave structure through the choice of the relaxation parameters. This flexibility can be exploited to handle robustly vacuum states, which is a well known difficulty of standard Roe's method, while maintaining Roe's low diffusivity. For the single-phase model positivity of flow height is rigorously preserved. For the two-phase model positivity of volume fractions in general is not ensured, and a suitable restriction on the CFL number might be needed. Nonetheless, numerical experiments suggest that the proposed two-phase flow solver efficiently models wet/dry fronts and vacuum formation for a large range of flow conditions.As a corollary of our study, we show that for single-phase shallow flow equations the relaxation solver is formally equivalent to the VFRoe solver with conservative variables of Gallouët and Masella [T. Gallouët, J.-M. Masella, Un schéma de Godunov approché C.R. Acad. Sci. Paris, Série I, 323 (1996) 77-84]. The relaxation interpretation allows establishing positivity conditions for this VFRoe method.

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“…It is interesting to note that the diagonal form of a Jin-Xin type relaxation system can be interpreted as a discrete velocity Boltzmann equation [1,4,33]. In the literature, lots of numerical studies have been reported in the context of both discrete Boltzmann and relaxation models, see [1,2,10,20,21,29,30,36,38,44] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

A Second Order Accurate Kinetic Relaxation Scheme for Inviscid Compressible Flows

Arun

Lukáčová-Medviďová

Prasad

et al. 2013

Notes on Numerical Fluid Mechanics and Multidisciplinary Design

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Abstract. In this paper we present a kinetic relaxation scheme for the Euler equations of gas dynamics in one space dimension. The method is easily applicable to solve any complex system of conservation laws. The numerical scheme is based on a relaxation approximation for conservation laws viewed as a discrete velocity model of the Boltzmann equation of kinetic theory. The discrete kinetic equation is solved by a splitting method consisting of a convection phase and a collision phase. The convection phase involves only the solution of linear transport equations and the collision phase instantaneously relaxes the distribution function to an equilibrium distribution. We prove that the first order accurate method is conservative, preserves the positivity of mass density and pressure and entropy stable. An anti-diffusive Chapman-Enskog distribution is used to derive a second order accurate method. The results of numerical experiments on some benchmark problems confirm the efficiency and robustness of the proposed scheme.

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Relaxation of Energy and Approximate Riemann Solvers for General Pressure Laws in Fluid Dynamics

Cited by 158 publications

References 15 publications

Finite volume simulation of cavitating flows

Finite volume simulation of cavitating flows

A Riemann solver for single-phase and two-phase shallow flow models based on relaxation. Relations with Roe and VFRoe solvers

A Second Order Accurate Kinetic Relaxation Scheme for Inviscid Compressible Flows

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