Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-26T11:07:37.349Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  03 April 2008

Dietmar Kröner ,
Philippe G. LeFloch  and
Mai-Duc Thanh
Dietmar Kröner
Affiliation:
Institute of Applied Mathematics, University of Freiburg, Hermann-Herder Str. 10, 79104 Freiburg, Germany. [email protected]
Philippe G. LeFloch
Affiliation:
Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique, Université de Paris VI, 4 Place Jussieu, 75252 Paris, France. [email protected]
Mai-Duc Thanh
Affiliation:
Department of Mathematics, International University, Quarter 6, Linh Trung Ward, Thu Duc District, Ho Chi Minh City, Vietnam. [email protected]
Get access

Abstract

We consider the Euler equations for compressible fluidsin a nozzle whose cross-section is variable and may contain discontinuities.We view these equations as a hyperbolic system in nonconservative formand 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.

Keywords

Euler equationsconservation lawshock wavenozzle flowsource termentropy solution.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 42 , Issue 3 , May 2008 , pp. 425 - 442
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Andrianov, N. and Warnecke, G., On the solution to the Riemann problem for the compressible duct flow. SIAM J. Appl. Math. 64 (2004) 878–901.
Audusse, E., Bouchut, F., Bristeau, M.-O., Klein, R. and Perthame, B., A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comp. 25 (2004) 2050–2065. CrossRef
Botchorishvili, R. and Pironneau, O., Finite volume schemes with equilibrium type discretization of source terms for scalar conservation laws. J. Comput. Phys. 187 (2003) 391–427. CrossRef
Botchorishvili, R., Perthame, B. and Vasseur, A., Equilibrium schemes for scalar conservation laws with stiff sources. Math. Comput. 72 (2003) 131–157. CrossRef
F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws, and well-balanced schemes for sources, Frontiers in Mathematics series. Birkhäuser (2004).
R. Courant and K.O. Friedrichs, Supersonic Flow and Shock Waves. John Wiley, New York (1948).
Dal Maso, G., LeFloch, P.G. and Murat, F., Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74 (1995) 483–548.
Goatin, P. and LeFloch, P.G., The Riemann problem for a class of resonant nonlinear systems of balance laws. Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004) 881–902. CrossRef
Gosse, L., A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms. Comp. Math. Appl. 39 (2000) 135–159. CrossRef
Greenberg, J.M. and Leroux, A.Y., A well-balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33 (1996) 1–16. CrossRef
A. Harten, P.D. Lax, C.D. Levermore and W.J. Morokoff, Convex entropies and hyperbolicity for general Euler equations. SIAM J. Numer. Anal. 35 2117–2127 (1998).
Isaacson, E. and Temple, B., Nonlinear resonance in systems of conservation laws. SIAM J. Appl. Math. 52 (1992) 1260–1278. CrossRef
Isaacson, E. and Temple, B., Convergence of the 2 x 2 Godunov method for a general resonant nonlinear balance law. SIAM J. Appl. Math. 55 (1995) 625–640. CrossRef
D. Kröner and M.D. Thanh, On the Model of Compressible Flows in a Nozzle: Mathematical Analysis and Numerical Methods, in Proc. 10th Intern. Conf. “Hyperbolic Problem: Theory, Numerics, and Applications”, Osaka (2004), Yokohama Publishers (2006) 117–124.
Kröner, D. and Thanh, M.D., Numerical solutions to compressible flows in a nozzle with variable cross-section. SIAM J. Numer. Anal. 43 (2006) 796–824. CrossRef
LeFloch, P.G., Entropy weak solutions to nonlinear hyperbolic systems under nonconservative form. Comm. Partial. Diff. Eq. 13 (1988) 669–727.
P.G. LeFloch, Shock waves for nonlinear hyperbolic systems in nonconservative form. Preprint 593, Institute Math. Appl., Minneapolis (1989).
P.G. LeFloch, Hyperbolic systems of conservation laws: The theory of classical and non-classical shock waves, Lectures in Mathematics. ETH Zürich, Birkäuser (2002).
LeFloch, P.G., Graph solutions of nonlinear hyperbolic systems. J. Hyper. Diff. Equ. 1 (2004) 243–289.
LeFloch, P.G. and Liu, T.-P., Existence theory for nonlinear hyperbolic systems in nonconservative form. Forum Math. 5 (1993) 261–280.
LeFloch, P.G. and Thanh, M.D., The Riemann problem for fluid flows in a nozzle with discontinuous cross-section. Comm. Math. Sci. 1 (2003) 763–797. CrossRef
LeFloch, P.G. and Thanh, M.D., The Riemann problem for the shallow water equations with discontinuous topography. Comm. Math. Sci. 5 (2007) 865–885. CrossRef
Marchesin, D. and Paes-Leme, P.J., Riemann, A problem in gas dynamics with bifurcation. Hyperbolic partial differential equations III. Comput. Math. Appl. (Part A) 12 (1986) 433–455. CrossRef
Tadmor, E., Skew selfadjoint form for systems of conservation laws. J. Math. Anal. Appl. 103 (1984) 428–442. CrossRef
Tadmor, E., A minimum entropy principle in the gas dynamics equations. Appl. Numer. Math. 2 (1986) 211–219. CrossRef