An approximate nonlinear projection scheme for a combustion model
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 3, p. 451-478

The paper deals with the numerical resolution of the convection-diffusion system which arises when modeling combustion for turbulent flow. The considered model is of compressible turbulent reacting type where the turbulence-chemistry interactions are governed by additional balance equations. The system of PDE's, that governs such a model, turns out to be in non-conservation form and usual numerical approaches grossly fail in the capture of viscous shock layers. Put in other words, classical finite volume methods induce large errors when approximated the convection-diffusion extracted system. To solve this difficulty, recent works propose a nonlinear projection scheme based on cancellation phenomenon of relevant dissipation rates of entropy. Unfortunately, such a property never holds in the present framework. The nonlinear projection procedures are thus extended.

DOI : https://doi.org/10.1051/m2an:2003037
Classification:  65M99,  76L05,  76M25
Keywords: hyperbolic systems in nonconservation form, finite volume methods, nonlinear projection method
@article{M2AN_2003__37_3_451_0,
     author = {Berthon, Christophe and Reignier, Didier},
     title = {An approximate nonlinear projection scheme for a combustion model},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {37},
     number = {3},
     year = {2003},
     pages = {451-478},
     doi = {10.1051/m2an:2003037},
     zbl = {1062.65102},
     mrnumber = {1994312},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2003__37_3_451_0}
}
An approximate nonlinear projection scheme for a combustion model. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 3, pp. 451-478. doi : 10.1051/m2an:2003037. http://www.numdam.org/item/M2AN_2003__37_3_451_0/

[1] R. Abgrall, An extension of Roe's upwind scheme to algebraic equilibrium real gas models. Comput. and Fluids 19 (1991) 171-182. | Zbl 0721.76061

[2] R.A. Baurle and S.S. Girimaji, An assumed PDF Turbulence-Chemistery closure with temperature-composition correlations. 37th Aerospace Sciences Meeting (1999).

[3] C. Berthon and F. Coquel, Travelling wave solutions of a convective diffusive system with first and second order terms in nonconservation form, Hyperbolic problems: theory, numerics, applications, vol. I, Zürich (1998) 47-54, Intern. Ser. Numer. Math. 129 Birkhäuser (1999). | Zbl 0934.35030

[4] C. Berthon and F. Coquel, About shock layers for compressible turbulent flow models, work in preparation, preprint MAB 01-29 2001 (http://www.math.u-bordeaux.fr/berthon). | MR 2247928

[5] C. Berthon and F. Coquel, Nonlinear projection methods for multi-entropies Navier-Stokes systems, Innovative methods for numerical solutions of partial differential equations, Arcachon (1998), World Sci. Publishing, River Edge (2002) 278-304. | Zbl 1078.76573

[6] C. Berthon, F. Coquel and P. Lefloch, Entropy dissipation measure and kinetic relation associated with nonconservative hyperbolic systems (in preparation).

[7] J.F. Colombeau, A.Y. Leroux, A. Noussair and B. Perrot, Microscopic profiles of shock waves and ambiguities in multiplications of distributions. SIAM J. Numer. Anal. 26 (1989) 871-883. | Zbl 0674.76049

[8] F. Coquel and P. Lefloch, Convergence of finite difference schemes for conservation laws in several space dimensions: a general theory. SIAM J. Numer. Anal. 30 (1993) 675-700. | Zbl 0781.65078

[9] F. Coquel and C. Marmignon, A Roe-type linearization for the Euler equations for weakly ionized multi-component and multi-temperature gas. Proceedings of the AIAA 12th CFD Conference, San Diego, USA (1995).

[10] F. Coquel and B. Perthame, Relaxation of energy and approximate Riemann solvers for general pressure laws in fluid dynamics. SIAM J. Numer. Anal. 35 (1998) 2223-2249. | Zbl 0960.76051

[11] G. Dal Maso, P. Lefloch and F. Murat, Definition and weak stability of a non conservative product. J. Math. Pures Appl. 74 (1995) 483-548. | Zbl 0853.35068

[12] A. Forestier, J.M. Herard and X. Louis, A Godunov type solver to compute turbulent compressible flows. C. R. Acad. Sci. Paris Sér. I Math. 324 (1997) 919-926. | Zbl 0881.76063

[13] E. Godlewski and P.A. Raviart, Hyperbolic systems of conservations laws. Springer, Appl. Math. Sci. 118 (1996). | MR 1410987 | Zbl 0860.65075

[14] A. Harten, P.D. Lax and B. Van Leer, On upstream differencing and Godunov type schemes for hyperbolic conservation laws. SIAM Rev. 25 (1983) 35-61. | Zbl 0565.65051

[15] T.Y. Hou and P.G. Lefloch, Why nonconservative schemes converge to wrong solutions: error analysis. Math. Comp. 62 (1994) 497-530. | Zbl 0809.65102

[16] L. Laborde, Modélisation et étude numérique de flamme de diffusion supersonique et subsonique en régime turbulent. Ph.D. thesis, Université Bordeaux I, France (1999).

[17] B. Larrouturou, How to preserve the mass fractions positivity when computing compressible multi-component flows. J. Comput. Phys. 95 (1991) 59-84. | Zbl 0725.76090

[18] B. Larrouturou and C. Olivier, On the numerical appproximation of the K-eps turbulence model for two dimensional compressible flows. INRIA report, No. 1526 (1991).

[19] P.G. Lefloch, Entropy weak solutions to nonlinear hyperbolic systems under non conservation form. Comm. Partial Differential Equations 13 (1988) 669-727. | Zbl 0683.35049

[20] B. Mohammadi and O. Pironneau, Analysis of the K-Epsilon Turbulence Model. Masson Eds., Rech. Math. Appl. (1994). | MR 1296252

[21] P.A. Raviart and L. Sainsaulieu, A nonconservative hyperbolic system modelling spray dynamics. Part 1. Solution of the Riemann problem. Math. Models Methods Appl. Sci. 5 (1995) 297-333. | Zbl 0837.76089

[22] P.L. Roe, Approximate Riemann solvers, parameter vectors and difference schemes. J. Comput. Phys. 43 (1981) 357-372. | Zbl 0474.65066

[23] L. Sainsaulieu, Travelling waves solutions of convection-diffusion systems whose convection terms are weakly nonconservative. SIAM J. Appl. Math. 55 (1995) 1552-1576. | Zbl 0841.35047

[24] E. Tadmor, A minimum entropy principle in the gas dynamics equations. Appl. Numer. Math. 2 (1986) 211-219. | Zbl 0625.76084