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http://dx.doi.org/10.4134/JKMS.2014.51.1.163

WELL-BALANCED ROE-TYPE NUMERICAL SCHEME FOR A MODEL OF TWO-PHASE COMPRESSIBLE FLOWS  

Thanh, Mai Duc (Department of Mathematics International University)
Publication Information
Journal of the Korean Mathematical Society / v.51, no.1, 2014 , pp. 163-187 More about this Journal
Abstract
We present a multi-stage Roe-type numerical scheme for a model of two-phase flows arisen from the modeling of deflagration-to-detonation transition in granular materials. The first stage in the construction of the scheme computes the volume fraction at every time step. The second stage deals with the nonconservative terms in the governing equations which produces states on both side of the contact wave at each node. In the third stage, a Roe matrix for the two-phase is used to apply on the states obtained from the second stage. This scheme is shown to capture stationary waves and preserves the positivity of the volume fractions. Finally, we present numerical tests which all indicate that the proposed scheme can give very good approximations to the exact solution.
Keywords
two-phase flow; balance law; nonconservative; source term; numerical approximation; well-balanced scheme; Roe-type scheme; shock wave; rarefaction wave; contact discontinuity;
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1 M. D. Thanh, A phase decomposition approach and the Riemann problem for a model of two-phase flows, preprint.
2 M. D. Thanh, The Riemann problem for a nonisentropic fluid in a nozzle with discontinuous cross-sectional area, SIAM J. Appl. Math. 69 (2009), no. 6, 1501-1519.   DOI   ScienceOn
3 M. D. Thanh, Exact solutions of a two-fluid model of two-phase compressible flows with gravity, Nonlinear Anal. Real World Appl. 13 (2012), no. 2, 987-998.   DOI   ScienceOn
4 M. D. Thanh, On a two-fluid model of two-phase compressible flows and its numerical approximation, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), no. 1, 195-211.   DOI   ScienceOn
5 M. D. Thanh and A. Izani Md. Ismail, A well-balanced scheme for a one-pressure model of two-phase flows, Phys. Scr. 79 (2009), no. 6, 065401, 7pp.   DOI   ScienceOn
6 M. D. Thanh, Md. Fazlul Karim, and A. Izani Md. Ismail, Well-balanced scheme for shallow water equations with arbitrary topography, Int. J. Dyn. Syst. Differ. Equ. 1 (2008), no. 3, 196-204.
7 M. D. Thanh, D. Kroner, and C. Chalons, A robust numerical method for approximating solutions of a model of two-phase flows and its properties, Appl. Math. Comput. 219 (2012), no. 1, 320-344.   DOI   ScienceOn
8 M. D. Thanh, D. Kroner, and N. T. Nam, Numerical approximation for a Baer-Nunziato model of two-phase flows, Appl. Numer. Math. 61 (2011), no. 5, 702-721.   DOI   ScienceOn
9 F. M. White, Fluid Mechanics, 7th ed. McGraw-Hill, 2010.
10 A. Ambroso, C. Chalons, F. Coquel, and T. Galie, Relaxation and numerical approximation of a two-fluid two-pressure diphasic model, Math. Model. Numer. Anal. 43 (2009), no. 6, 1063-1097.   DOI   ScienceOn
11 N. Andrianov and G.Warnecke, The Riemann problem for the Baer-Nunziato two-phase flow model, J. Comput. Phys. 195 (2004), no. 2, 434-464.   DOI   ScienceOn
12 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. Comput. 25 (2004), no. 6, 2050-2065.   DOI   ScienceOn
13 M. R. Baer and J. W. Nunziato, A two-phase mixture theory for the de agration-to-detonation transition (ddt) in reactive granular materials, Int. J. Multiphase Flow 12 (1986), no. 6, 861-889.   DOI   ScienceOn
14 R. Botchorishvili, B. Perthame, and A. Vasseur, Equilibrium schemes for scalar conservation laws with stiff sources, Math. Compu. 72 (2003), no. 241, 131-157.
15 R. Botchorishvili and O. Pironneau, Finite volume schemes with equilibrium type discretization of source terms for scalar conservation laws, J. Comput. Phys. 187 (2003), no. 2, 391-427.   DOI   ScienceOn
16 J. B. Bzil, R. Menikoff, S. F. Son, A. K. Kapila, and D. S. Steward, Two-phase modelling of a de agration-to-detonation transition in granular materials: A critical examination of modelling issues, Phys. Fluids 11 (1999), no. 2, 378-402.   DOI   ScienceOn
17 A. Chinnayya, A.-Y. LeRoux, and N. Seguin, A well-balanced numerical scheme for the approximation of the shallow water equations with topography: the resonance phenomenon, Int. J. Finite Vol. 1 (2004), no. 1, 33 pp.
18 T. Gallouet, J.-M. Herard, and N. Seguin, Numerical modeling of two-phase flows using the two-fluid two-pressure approach, Math. Models Methods Appl. Sci. 14 (2004), no. 5, 663-700.   DOI   ScienceOn
19 G. Dal Maso, P. G. LeFloch, and F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl. (9) 74 (1995), no. 6, 483-548.
20 P. Embid and M. Baer, Mathematical analysis of a two-phase continuum mixture theory, Contin. Mech. Thermodyn. 4 (1992), no. 4, 279-312.   DOI
21 P. Goatin and P. G. LeFloch, The Riemann problem for a class of resonant hyperbolic systems of balance laws, Ann. Inst. H. Poincare Anal. Non Lineaire 21 (2004), no. 6881-902.
22 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), no. 1, 1-16.   DOI   ScienceOn
23 J. M. Greenberg, A. Y. Leroux, R. Baraille, and A. Noussair, Analysis and approximation of conservation laws with source terms, SIAM J. Numer. Anal. 34 (1997), no. 5, 1980-2007.   DOI   ScienceOn
24 S. Jin and X. Wen, An efficient method for computing hyperbolic systems with geometrical source terms having concentrations, J. Comput. Math. 22 (2004), no. 2, 230-249.
25 S. Karni and G. Hernandez-Duenas, A hybrid algorithm for the Baer-Nunziato model using the Riemann invariants, J. Sci. Comput. 45 (2010), no. 1-3, 382-403.   DOI
26 B. L. Keyfitz, R. Sander, and M. Sever, Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow, Discrete Contin. Dyn. Syst. Ser. B 3 (2003), no. 4, 541-563.   DOI
27 P. G. LeFloch and M. D. Thanh, The Riemann problem for fluid flows in a nozzle with discontinuous cross-section, Commun. Math. Sci. 1 (2003), no. 4, 763-797.   DOI
28 D. Kroner, P. G. LeFloch, and M. D. Thanh, The minimum entropy principle for compressible fluid flows in a nozzle with discontinuous cross-section, Math. Model. Numer. Anal. 42 (2008), no. 3, 425-442.   DOI   ScienceOn
29 D. Kroner and M. D. Thanh, Numerical solutions to compressible flows in a nozzle with variable cross-section, SIAM J. Numer. Anal. 43 (2005), no. 2, 796-824.   DOI   ScienceOn
30 M.-H. Lallemand and R. Saurel, Pressure relaxation procedures for multiphase compressible flows, INRIA Report (2000), No. 4038.
31 P. G. LeFloch and M. D. Thanh, The Riemann problem for the shallow water equations with discontinuous topography, Commun. Math. Sci. 5 (2007), no. 4, 865-885.   DOI
32 P. G. LeFloch and M. D. Thanh, A Godunov-type method for the shallow water equations with discontinuous topography in the resonant regime, J. Comput. Phys. 230 (2011), no. 20, 7631-7660.   DOI   ScienceOn
33 S. T. Munkejord, Comparison of Roe-type methods for solving the two-fluid model with and without pressure relaxation, Computers & Fluids 36 (2007), no. 6, 1061-1080.   DOI   ScienceOn
34 R. Saurel and R. Abgrall, A multiphase Godunov method for compressible multifluid and multiphase flows, J. Comput. Phys. 150 (1999), no. 2, 425-467.   DOI   ScienceOn
35 D. W. Schwendeman, C. W. Wahle, and A. K. Kapila, The Riemann problem and a high-resolution godunov method for a model of compressible two-phase flow, J. Comput. Phys. 212 (2006), no. 2, 490-526.   DOI   ScienceOn