Geodesic
Exact and approximate Riemann solvers for compressible two-phase flows
Matematičeskoe modelirovanie, Tome 28 (2016) no. 12, pp. 33-55.

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A numerical method for solving the two-phase hydrodynamics equations that describe the flow of dispersed solid and gas mixture is considered. The Godunov method is applied to approximate numerical fluxes with implementing solutions to the Riemann problem. The formulations of these problems for the solid and gas phases are given, their exact analytical solutions are described and possible simplified approximate solutions are discussed. The obtained theoretical results are applied to construction of the discrete model which leads to extension of well-known Godunov-type and Rusanov-type methods to the system of Baer–Nunziato equations for non-equilibrium twophase flows. Numerical results concern the method verification on generalized Sod problems with analytical self-similar solutions of two-phase equations.
Keywords: two-phase flow, Riemann problem, numerical methods of Godunov and Rusanov.
Mots-clés : non-conservative Euler equations 
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     title = {Exact and approximate {Riemann} solvers for compressible two-phase flows},
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     url = {http://geodesic.mathdoc.fr/item/MM_2016_28_12_a2/}
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Igor Menshov. Exact and approximate Riemann solvers for compressible two-phase flows. Matematičeskoe modelirovanie, Tome 28 (2016) no. 12, pp. 33-55. http://geodesic.mathdoc.fr/item/MM_2016_28_12_a2/

[1] R. I. Nigmatulin, Dinamika mnogofaznykh sred, v. 1, Nauka, M., 1987, 464 pp.

[2] S. Clain, D. Rochette, “First- and second-order finite volume methods for the one-dimensional nonconservative Euler system”, J. Comp. Phys., 228 (2009), 8214–8248 | DOI | MR | Zbl

[3] N. Andrianov, G. Warnecke, “The Riemann problem for the Baer-Nunziato two-phase flow model”, J. Comput. Phys., 195 (2004), 434–464 | DOI | MR | Zbl

[4] M.E. Vazquez-Cendon, “Improved treatment of source terms in upwind schemes for the shallow-water equations in channels with irregular geometry”, J. Comp. Phys., 148 (1999), 497–525 | DOI | MR

[5] R. Saurel, R. Abgrall, “A multiphase Godunov method for compressible multifluid and multiphase flows”, J. Comput. Phys., 150 (1999), 425–467 | DOI | MR | Zbl

[6] N. Andrianov, G. Warnecke, “On the solution to the Riemann problem for the compressible duct flow”, SIAM J. Appl. Math., 64:3 (2004), 878–901 | DOI | MR | Zbl

[7] G. Dal Maso, P. G. Le Floch, F. Murat, “Definition and weak stability of a non-conservative product”, J. Math. Pures. Appl., 74:6 (1995), 483–548 | MR | Zbl

[8] L. Gosse, “A well-balanced flux-vector splitting scheme designed for hyperbolic system of conservation laws with source terms”, Comput. Math. Appl., 39 (2000), 135–159 | DOI | MR | Zbl

[9] V. V. Rusanov, “Raschet vzaimodeistviia nestatsionarnykh udarnykh voln c prepiatstviiami”, Zh. vychisl. matem. i matem. fiz., 1:2 (1961), 267–279

[10] R. Abgrall, R. Saurel, “Discrete equations for physical and numerical compressible multiphase mixtures”, J. Comput. Phys., 186 (2003), 361–396 | DOI | MR | Zbl

[11] J. M. Greenberg, A. Y. Leroux, “A well-balanced scheme for the numerical processing of source terms in hyperbolic equations”, SIAM J. Numer. Anal., 3:1 (1996), 1–16 | DOI | MR

[12] A. Bermudez, M. E. Vazquez, “Upwind methods for hyperbolic conservation laws with source term”, Comput. Fluid, 23 (1994), 1049–1071 | DOI | MR | Zbl

[13] C. Pares, M. Castro, “On the well-balance property of Roe's method for nonconservative hyperbolic systems. Applications to shallow-water systems”, ESAIM: Math. Model. Numer. Anal., 38:5 (2004), 821–852 | DOI | MR | Zbl

[14] D. W. Schwendeman, C. W. Wahle, A. K. Kapila, “The Riemann problem and high-resolution Godunov method for a model of compressible two-phase flow”, J. Comp. Phys., 212 (2006), 490–526 | DOI | MR | Zbl

[15] B. W. Asay, S. F. Son, J. B. Bdzil, “The role of gas permeation in convective burning”, Int. J. Multiphase Flow, 23:5 (1996), 923–952 | DOI

[16] S. A. Tokareva, E. F. Toro, “HLLC-type Riemann solver for the Baer–Nunziato equations of compressible two-phase flow”, J. Comp. Phys., 229 (2010), 3573–3604 | DOI | MR | Zbl

[17] S. K. Godunov, “Raznostnyi metod chislennogo rascheta razryvnykh reshenii uravnenii gidrodinamiki”, Matem. sb., 47:3 (1959), 271–306 | Zbl

[18] J. Bell, P. Colella, J. Trangenstein, “Higher Order Godunov Methods for General Systems of Hyperbolic Conservation Laws”, J. Comput. Phys., 82 (1989), 362–397 | DOI | MR | Zbl

[19] A. V. Rodionov, “Monotonnaya skhema vtorogo poryadka approksimatsii dlya marshevykh raschetov neravnovesnykh potokov”, ZhVMiMF, 27:4 (1987), 585–593 | Zbl

[20] I. Menshov, A. Serezhkin, “Modeling Non-Equilibrium Two-Phase Flow in Elastic-Plastic Porous Solids”, Proceedings IACM-ECCOMAS 2014, Electronic book, 2014, 1–12

[21] T. Gallouët, J. M. Hérard, N. Seguin, “Some approximate Godunov schemes to compute shallowwater equations with topography”, Computers Fluids, 32:4 (2003), 479–513 | DOI | MR | Zbl

[22] S. K. Godunov, A. V. Zabrodin, M. Ia. Ivanov, A. N. Kraiko, G. P. Prokopov, Chislennoe reshenie mnogomernykh zadach gazovoi dinamiki, Nauka, M., 1976, 400 pp. | MR

[23] J. Dukowicz, “A General, Non-Iterative Riemann Solver for Godunov's Method”, J. Comput. Phys., 61 (1985), 119–137 | DOI | MR | Zbl