The minimum entropy principle for compressible fluid flows in a nozzle with discontinuous cross-section
ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 3, pp. 425-442.

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.

DOI : 10.1051/m2an:2008011
Classification : 35L65, 76N10, 76L05
Mots clés : Euler equations, conservation law, shock wave, nozzle flow, source term, entropy solution
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     title = {The minimum entropy principle for compressible fluid flows in a nozzle with discontinuous cross-section},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
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Kröner, Dietmar; Lefloch, Philippe G.; Thanh, Mai-Duc. The minimum entropy principle for compressible fluid flows in a nozzle with discontinuous cross-section. ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 3, pp. 425-442. doi : 10.1051/m2an:2008011. http://www.numdam.org/articles/10.1051/m2an:2008011/

[1] N. Andrianov and G. Warnecke, On the solution to the Riemann problem for the compressible duct flow. SIAM J. Appl. Math. 64 (2004) 878-901. | MR | Zbl

[2] E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein and B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comp. 25 (2004) 2050-2065. | MR | Zbl

[3] R. Botchorishvili and O. Pironneau, Finite volume schemes with equilibrium type discretization of source terms for scalar conservation laws. J. Comput. Phys. 187 (2003) 391-427. | MR | Zbl

[4] R. Botchorishvili, B. Perthame and A. Vasseur, Equilibrium schemes for scalar conservation laws with stiff sources. Math. Comput. 72 (2003) 131-157. | MR | Zbl

[5] F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws, and well-balanced schemes for sources, Frontiers in Mathematics series. Birkhäuser (2004). | MR | Zbl

[6] R. Courant and K.O. Friedrichs, Supersonic Flow and Shock Waves. John Wiley, New York (1948). | MR | Zbl

[7] G. Dal Maso, P.G. Lefloch and F. Murat, Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74 (1995) 483-548. | MR | Zbl

[8] P. Goatin and P.G. Lefloch, The Riemann problem for a class of resonant nonlinear systems of balance laws. Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004) 881-902. | Numdam | MR | Zbl

[9] L. Gosse, A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms. Comp. Math. Appl. 39 (2000) 135-159. | MR | Zbl

[10] J.M. Greenberg and A.Y. Leroux, A well-balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33 (1996) 1-16. | MR | Zbl

[11] 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). | MR | Zbl

[12] E. Isaacson and B. Temple, Nonlinear resonance in systems of conservation laws. SIAM J. Appl. Math. 52 (1992) 1260-1278. | MR | Zbl

[13] E. Isaacson and B. Temple, Convergence of the 2×2 Godunov method for a general resonant nonlinear balance law. SIAM J. Appl. Math. 55 (1995) 625-640. | MR | Zbl

[14] 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. | Zbl

[15] D. Kröner and M.D. Thanh, Numerical solutions to compressible flows in a nozzle with variable cross-section. SIAM J. Numer. Anal. 43 (2006) 796-824. | MR | Zbl

[16] P.G. Lefloch, Entropy weak solutions to nonlinear hyperbolic systems under nonconservative form. Comm. Partial. Diff. Eq. 13 (1988) 669-727. | MR | Zbl

[17] P.G. Lefloch, Shock waves for nonlinear hyperbolic systems in nonconservative form. Preprint 593, Institute Math. Appl., Minneapolis (1989).

[18] P.G. Lefloch, Hyperbolic systems of conservation laws: The theory of classical and non-classical shock waves, Lectures in Mathematics. ETH Zürich, Birkäuser (2002). | MR | Zbl

[19] P.G. Lefloch, Graph solutions of nonlinear hyperbolic systems. J. Hyper. Diff. Equ. 1 (2004) 243-289. | MR | Zbl

[20] P.G. Lefloch and T.-P. Liu, Existence theory for nonlinear hyperbolic systems in nonconservative form. Forum Math. 5 (1993) 261-280. | MR | Zbl

[21] P.G. Lefloch and M.D. Thanh, The Riemann problem for fluid flows in a nozzle with discontinuous cross-section. Comm. Math. Sci. 1 (2003) 763-797. | MR | Zbl

[22] P.G. Lefloch and M.D. Thanh, The Riemann problem for the shallow water equations with discontinuous topography. Comm. Math. Sci. 5 (2007) 865-885. | MR | Zbl

[23] D. Marchesin and P.J. Paes-Leme, A Riemann problem in gas dynamics with bifurcation. Hyperbolic partial differential equations III. Comput. Math. Appl. (Part A) 12 (1986) 433-455. | MR | Zbl

[24] E. Tadmor, Skew selfadjoint form for systems of conservation laws. J. Math. Anal. Appl. 103 (1984) 428-442. | MR | Zbl

[25] E. Tadmor, A minimum entropy principle in the gas dynamics equations. Appl. Numer. Math. 2 (1986) 211-219. | MR | Zbl

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