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

  • Dietmar Kröner (a1), Philippe G. LeFloch (a2) and Mai-Duc Thanh (a3)

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.

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[1] Andrianov, N. and Warnecke, G., On the solution to the Riemann problem for the compressible duct flow. SIAM J. Appl. Math. 64 (2004) 878901.
[2] 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) 20502065.
[3] Botchorishvili, R. and Pironneau, O., Finite volume schemes with equilibrium type discretization of source terms for scalar conservation laws. J. Comput. Phys. 187 (2003) 391427.
[4] Botchorishvili, R., Perthame, B. and Vasseur, A., Equilibrium schemes for scalar conservation laws with stiff sources. Math. Comput. 72 (2003) 131157.
[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).
[6] R. Courant and K.O. Friedrichs, Supersonic Flow and Shock Waves. John Wiley, New York (1948).
[7] Dal Maso, G., LeFloch, P.G. and Murat, F., Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74 (1995) 483548.
[8] 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) 881902.
[9] Gosse, L., A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms. Comp. Math. Appl. 39 (2000) 135159.
[10] 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) 116.
[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).
[12] Isaacson, E. and Temple, B., Nonlinear resonance in systems of conservation laws. SIAM J. Appl. Math. 52 (1992) 12601278.
[13] 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) 625640.
[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.
[15] Kröner, D. and Thanh, M.D., Numerical solutions to compressible flows in a nozzle with variable cross-section. SIAM J. Numer. Anal. 43 (2006) 796824.
[16] LeFloch, P.G., Entropy weak solutions to nonlinear hyperbolic systems under nonconservative form. Comm. Partial. Diff. Eq. 13 (1988) 669727.
[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).
[19] LeFloch, P.G., Graph solutions of nonlinear hyperbolic systems. J. Hyper. Diff. Equ. 1 (2004) 243289.
[20] LeFloch, P.G. and Liu, T.-P., Existence theory for nonlinear hyperbolic systems in nonconservative form. Forum Math. 5 (1993) 261280.
[21] 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) 763797.
[22] LeFloch, P.G. and Thanh, M.D., The Riemann problem for the shallow water equations with discontinuous topography. Comm. Math. Sci. 5 (2007) 865885.
[23] 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) 433455.
[24] Tadmor, E., Skew selfadjoint form for systems of conservation laws. J. Math. Anal. Appl. 103 (1984) 428442.
[25] Tadmor, E., A minimum entropy principle in the gas dynamics equations. Appl. Numer. Math. 2 (1986) 211219.

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

  • Dietmar Kröner (a1), Philippe G. LeFloch (a2) and Mai-Duc Thanh (a3)

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