The Riemann Problem for Fluid Flows in a Nozzle with Discontinuous Cross-Section

LeFloch, Philippe G.; Thanh, Mai Duc

doi:10.4310/cms.2003.v1.n4.a6

Cited by 103 publications

(102 citation statements)

References 6 publications

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“…Examples of non-conservative hyperbolic systems include shallow water equations with bottom topography [3], gas flows in a duct [15], multi-layer shallow water equations [1,2,4] and multi-phase flows [26].…”

Section: Systems In Non-conservative Formmentioning

confidence: 99%

Accurate numerical discretizations of non-conservative hyperbolic systems

Fjordholm

Mishra

2011

ESAIM: M2AN

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Abstract. We present an alternative framework for designing efficient numerical schemes for nonconservative hyperbolic systems. This approach is based on the design of entropy conservative discretizations and suitable numerical diffusion operators that mimic the effect of underlying viscous mechanisms. This approach is illustrated by considering two model non-conservative systems: Lagrangian gas dynamics in non-conservative form and a form of isothermal Euler equations. Numerical experiments demonstrating the robustness of this approach are presented.Mathematics Subject Classification. 65M06, 35L65.

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Section: Systems In Non-conservative Formmentioning

confidence: 99%

Accurate numerical discretizations of non-conservative hyperbolic systems

Fjordholm

Mishra

2011

ESAIM: M2AN

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“…For some forms of the function G(x, W ) the Riemann solution R s can be exactly computed (see [3] for the Shallow Water equations, [43] for the isentropic Euler equations and [5] for the full non-isentropic Euler equations).…”

Section: Construction Of Srnh Schemementioning

confidence: 99%

A sign matrix based scheme for non-homogeneous PDE’s with an analysis of the convergence stagnation phenomenon

Sahmim¹,

Benkhaldoun

Alcrudo

2007

Journal of Computational Physics

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To cite this version:Slah Sahmim, Fayssal Benkhaldoun, Francisco Alcrudo. A sign matrix based scheme for nonhomogeneous PDE's with an analysis of the convergence stagnation phenomenon. Journal of Computational Physics, Elsevier, 2007, 226 (2) AbstractThis work is devoted to the analysis of a finite volume method recently proposed for the numerical computation of a class of non homogenous systems of partial differencial equations of interest in fluid dynamics. The stability analysis of the proposed scheme leads to the introduction of the sign matrix of the flux jacobian. It appears that this formulation is equivalent to the VFRoe scheme introduced in the homogeneous case and has a natural extension here to non homogeneous systems. Comparative numerical experiments for the Shallow Water and Euler equations with source terms, and a model problem of two phase flow (Ransom faucet) are presented to validate the scheme. The numerical results present a convergence stagnation phenomenon for certain forms of the source term, notably when it is singular. Convergence stagnation has been also shown in the past for other numerical schemes. This issue is addressed in a specific section where an explanation is given with the help of a linear model equation, and a cure is demonstrated.

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“…Hence, the third equation of (4.5) implies that the component a can be expressed as a function a = a(ρ) of the variable ρ along this curve. We then postulate the following admissibility criterion (see [12,21] …”

Section: Equilibrium States and Admissibility Criterionmentioning

confidence: 99%

“…In consequence, at least within the regime where the system is strictly hyperbolic, the theory of such systems developed by LeFloch and co-authors (see [7] and also [16][17][18][19][20][21][22]) applies, and provide the existence of entropy solutions to the Riemann problem (a single discontinuity separating two constant states as an initial data), as well as to the Cauchy problem (for solution with sufficiently small total variation). More recently, LeFloch and Thanh [21,22] solved the Riemann problem for arbitrary data, including the regime where the system fails to be globally strict hyperbolicity (i.e., the resonant case).…”

Section: Introductionmentioning

confidence: 99%

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The minimum entropy principle for compressible fluid flows in a nozzle with discontinuous cross-section

2008

Self Cite

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Abstract. We consider the Euler equations for compressible fluids in a nozzle whose cross-section is variable and may contain discontinuities. We view these equations as a hyperbolic system in nonconservative form and investigate weak solutions in the sense of Dal Maso, LeFloch and Murat [J. Math. Pures Appl. 74 (1995) 483-548]. Observing that the entropy equality has a fully conservative form, we derive a minimum entropy principle satisfied by entropy solutions. We then establish the stability of a class of numerical approximations for this system. Mathematics Subject Classification. 35L65, 76N10, 76L05.

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The Riemann Problem for Fluid Flows in a Nozzle with Discontinuous Cross-Section

Cited by 103 publications

References 6 publications

Accurate numerical discretizations of non-conservative hyperbolic systems

Accurate numerical discretizations of non-conservative hyperbolic systems

A sign matrix based scheme for non-homogeneous PDE’s with an analysis of the convergence stagnation phenomenon

The minimum entropy principle for compressible fluid flows in a nozzle with discontinuous cross-section

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