A robust and entropy-satisfying numerical scheme for fluid flows in discontinuous nozzles

Coquel, Fré́dé́ric; Saleh, Khaled; Seguin, Nicolas

doi:10.1142/s0218202514500158

Cited by 15 publications

(23 citation statements)

References 27 publications

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“…In the equality case, the solution is said to be energy-preserving. System (4.33) is nothing but the relaxation system introduced in [17] for the approximation of nozzle flows, and for which the associated Riemann problem has been fully resolved. Hence, we actually calculate a solution W(y, t) of the Riemann problem associated with system (4.33), and the solution for the original Riemann problem (4.20)−(4.21) is obtained by W(x, t) = W(x− u * 2 t, t), and by adding u * 2 to the velocities w 1 .…”

Section: A Convenient Change Of Variablesmentioning

confidence: 99%

“…For the resolution, we only consider solutions with the subsonic wave ordering u * 2 < u * 1 since the other possible wave orderings can be obtained by the Galilean invariance of the equations. This ordering for (4.20) corresponds to the wave ordering w 1 − a 1 τ 1 < 0 < w 1 < w 1 + a 1 τ 1 for (4.33), which in [17] is referred to as the < 1, 2 > wave configuration:…”

Section: A Convenient Change Of Variablesmentioning

confidence: 99%

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A robust entropy−satisfying finite volume scheme for the isentropic Baer−Nunziato model

Coquel

Hérard²,

Saleh

et al. 2013

ESAIM: M2AN

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We construct an approximate Riemann solver for the isentropic Baer−Nunziato two-phase flow model, that is able to cope with arbitrarily small values of the statistical phase fractions. The solver relies on a relaxation approximation of the model for which the Riemann problem is exactly solved for subsonic relative speeds. In an original manner, the Riemann solutions to the linearly degenerate relaxation system are allowed to dissipate the total energy in the vanishing phase regimes, thereby enforcing the robustness and stability of the method in the limits of small phase fractions. The scheme is proved to satisfy a discrete entropy inequality and to preserve positive values of the statistical fractions and densities. The numerical simulations show a much higher precision and a more reduced computational cost (for comparable accuracy) than standard numerical schemes used in the nuclear industry. Finally, two test-cases assess the good behavior of the scheme when approximating vanishing phase solutions.

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Section: A Convenient Change Of Variablesmentioning

confidence: 99%

Section: A Convenient Change Of Variablesmentioning

confidence: 99%

A robust entropy−satisfying finite volume scheme for the isentropic Baer−Nunziato model

Coquel

Hérard²,

Saleh

et al. 2013

ESAIM: M2AN

Self Cite

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show abstract

“…Existence and stability results for the shallow water system have been established in [27,21,23,28]. Concerning approximation, many schemes have been proposed, see for example [25,4,3,6,12,7,11,19,20]. The hydrostatic reconstruction scheme and its variants [4,24,18,17,16,15] is often used, and it is the subject of the present paper.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…According to the property (A. 19) we can now define a path v(t) ∈ U m , for 0 ≤ t ≤ 1, connecting the two states U 1 , U 2 satisfying (A.18), by…”

Section: )mentioning

confidence: 99%

Convergence of the kinetic hydrostatic reconstruction scheme for the Saint Venant system with topography

Bouchut

Lhébrard

2020

Math. Comp.

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We prove the convergence of the hydrostatic reconstruction scheme with kinetic numerical flux for the Saint Venant system with continuous topography with locally integrable derivative. We use a recently derived fully discrete sharp entropy inequality with dissipation, that enables us to establish an estimate in the inverse of the square root of the space increment ∆x of the L 2 norm of the gradient of approximate solutions. By Diperna's method we conclude the strong convergence towards bounded weak entropy solutions.

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“…In particular, condition (F4) may be hard to obtain. One may mention some of them: the non-conservative Godunov scheme [28], a modified kinetic scheme [35], Suliciu's relaxation method [5,13,4], entropy-stable schemes [18]. .…”

Section: An Example Of Well-balanced Schemementioning

confidence: 99%

Stability of stationary solutions of singular systems of balance laws

Seguin¹

2019

Confluentes Mathematici

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The stability of stationary solutions of first-order systems of PDE's are considered. They may include some singular geometric terms, leading to discontinuous flux and non-conservative products. Based on several examples in Fluid Mechanics, we assume that these systems are endowed with a partially convex entropy. We first construct an associated relative entropy which allows to compare two states which share the same geometric data. This way, we are able to prove the stability of some stationary states within entropy weak solutions. This result applies for instance to the shallow-water equations with bathymetry. Besides, this relative entropy can be used to study finite volume schemes which are entropystable and well-balanced, and due to the numerical dissipation inherent to these methods, asymptotic stability of discrete stationary solutions is obtained. This analysis does not make us of any specific definition of the non-conservative products, applies to non-strictly hyperbolic systems, and is fully multidimensional with unstructured meshes for the numerical methods.

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A robust and entropy-satisfying numerical scheme for fluid flows in discontinuous nozzles

Cited by 15 publications

References 27 publications

A robust entropy−satisfying finite volume scheme for the isentropic Baer−Nunziato model

A robust entropy−satisfying finite volume scheme for the isentropic Baer−Nunziato model

Convergence of the kinetic hydrostatic reconstruction scheme for the Saint Venant system with topography

Stability of stationary solutions of singular systems of balance laws

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