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Alina Chertock; Shumo Cui; Alexander Kurganov
Hybrid Finite-Volume-Particle Method for Dusty Gas Flows
SMAI-Journal of computational mathematics, 3 (2017), p. 139-180, doi: 10.5802/smai-jcm.23
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Class. Math.: 65M08, 76M12, 76M28, 86-08, 76M25, 35L65
Keywords: Two-phase dusty gas flow model, three-dimensional axial symmetry, compressible Euler equations, pressureless gas dynamics, finite-volume-particle methods, central-upwind schemes, sticky particle methods, operator splitting methods

Abstract

We first study the one-dimensional dusty gas flow modeled by the two-phase system composed of a gaseous carrier (gas phase) and a particulate suspended phase (dust phase). The gas phase is modeled by the compressible Euler equations of gas dynamics and the dust phase is modeled by the pressureless gas dynamics equations. These two sets of conservation laws are coupled through source terms that model momentum and heat transfers between the phases. When an Eulerian method is adopted for this model, one can notice the obtained numerical results are typically significantly affected by numerical diffusion. This phenomenon occurs since the pressureless gas equations are nonstrictly hyperbolic and have degenerate structure in which singular delta shocks are formed, and these strong singularities are vulnerable to the numerical diffusion.

We introduce a low dissipative hybrid finite-volume-particle method in which the compressible Euler equations for the gas phase are solved by a central-upwind scheme, while the pressureless gas dynamics equations for the dust phase are solved by a sticky particle method. The obtained numerical results demonstrate that our hybrid method provides a sharp resolution even when a relatively small number of particle is used.

We then extend the hybrid finite-volume-particle method to the three-dimensional dusty gas flows with axial symmetry. In the studied model, gravitational effects are taken into account. This brings an additional level of complexity to the development of the finite-volume-particle method since a delicate balance between the flux and gravitational source terms should be respected at the discrete level. We test the proposed method on a number of numerical examples including the one that models volcanic eruptions.

Bibliography

[1] N. Botta, R. Klein, S. Langenberg & S. Lützenkirchen, “Well balanced finite volume methods for nearly hydrostatic flows”, J. Comput. Phys. 196 (2004) no. 2, p. 539-565 Article |  MR 2054350
[2] F. Bouchut, On zero pressure gas dynamics, Advances in kinetic theory and computing, Ser. Adv. Math. Appl. Sci. 22, World Sci. Publ., River Edge, NJ, 1994, p. 171–190
[3] F. Bouchut & F. James, “Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness”, Comm. Partial Differential Equations 24 (1999) no. 11-12, p. 2173-2189 Article
[4] F. Bouchut, S. Jin & X. Li, “Numerical approximations of pressureless and isothermal gas dynamics”, SIAM J. Numer. Anal. 41 (2003), p. 135-158 Article
[5] Y. Brenier & E. Grenier, “Sticky particles and scalar conservation laws”, SIAM J. Numer. Anal. 35 (1998), p. 2317-2328 Article
[6] S. Carcano, L. Bonaventura, T. Esposti Ongaro & A. Neri, “A semi-implicit, second order accurate numerical model for multiphase underexpanded volcanic jets”, Geosci. Model Dev. Discuss. 6 (2013) no. 1, p. 399-452 Article
[7] P. Chandrashekar & C. Klingenberg, “A second order well-balanced finite volume scheme for Euler equations with gravity”, SIAM J. Sci. Comput. 37 (2015) no. 3, p. B382-B402 Article
[8] G.-Q. Chen & H. Liu, “Formation of $\delta $-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids”, SIAM J. Math. Anal. 34 (2003), p. 925-938 Article
[9] A. Chertock, S. Cui, A. Kurganov, Ş. N. Özcan & E. Tadmor, “Well-balanced central-upwind schemes for the Euler equations with gravitation”, Submitted
[10] A. Chertock, S. Cui, A. Kurganov & T. Wu, “Well-balanced positivity preserving central-upwind scheme for the shallow water system with friction terms”, Internat. J. Numer. Meth. Fluids 78 (2015), p. 355-383 Article
[11] A. Chertock & A. Kurganov, “On a hybrid finite-volume particle method”, M2AN Math. Model. Numer. Anal 38 (2004), p. 1071-1091 Article
[12] A. Chertock & A. Kurganov, “On a practical implementation of particle methods”, Appl. Numer. Math. 56 (2006), p. 1418-1431 Article
[13] A. Chertock, A. Kurganov & G. Petrova, “Finite-volume-particle methods for models of transport of pollutant in shallow water”, J. Sci. Comput. 27 (2006), p. 189-199 Article
[14] A. Chertock, A. Kurganov & Yu. Rykov, “A new sticky particle method for pressureless gas dynamics”, SIAM J. Numer. Anal. 45 (2007), p. 2408-2441 Article
[15] G.-H. Cottet & P. D. Koumoutsakos, Vortex methods, Cambridge University Press, Cambridge, 2000
[16] S. Dartevelle, W. Rose, J. Stix, K. Kelfoun & J.W. Vallance, “Numerical modeling of geophysical granular flows: 2. Computer simulations of plinian clouds and pyroclastic flows and surges”, Geochem. Geophys. Geosyst. 5 (2004) no. 8 Article
[17] V. Desveaux, M. Zenk, C. Berthon & C. Klingenberg, A well-balanced scheme for the Euler equation with a gravitational potential, Finite volumes for complex applications. VII. Methods and theoretical aspects, Springer Proc. Math. Stat. 77, Springer, Cham, 2014, p. 217–226
[18] F. Dobran, A. Neri & G. Macedonio, “Numerical simulation of collapsing volcanic columns”, J. Geophys. Res. 98 (1993), p. 4231-4259 Article
[19] J. Dufek & G. W. Bergantz, “Dynamics and deposits generated by the Kos Plateau Tuff eruption: Controls of basal particle loss on pyroclastic flow transport”, Geochem. Geophys. Geosyst. 8 (2007) no. 12 Article
[20] W. E, Yu. G. Rykov & Ya. G. Sinai, “Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics”, Comm. Math. Phys. 177 (1996), p. 349-380 Article
[21] B. Einfeld, “On Godunov-type methods for gas dynamics”, SIAM J. Numer. Anal. 25 (1988), p. 294-318 Article
[22] P. Glaister, “Flux difference splitting for the Euler equations with axial symmetry”, J. Engrg. Math. 22 (1988) no. 2, p. 107-121 Article
[23] S. Gottlieb, D. Ketcheson & C.-W. Shu, Strong stability preserving Runge-Kutta and multistep time discretizations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011
[24] S. Gottlieb, C.-W. Shu & E. Tadmor, “Strong stability-preserving high-order time discretization methods”, SIAM Rev. 43 (2001), p. 89-112 Article
[25] M. Gurris, D. Kuzmin & S. Turek, “Finite element simulation of compressible particle-laden gas flows”, J. Comput. Appl. Math. 233 (2010) no. 12, p. 3121-3129 Article
[26] S. Hank, R. Saurel & O. Le Metayer, “A hyperbolic Eulerian model for dilute two-phase suspensions”, Journal of Modern Physics 2 (2011), p. 997-1011 Article
[27] A. Harlow & A. A. Amsden, “Numerical calculation of multiphase fluid flow”, J. Comput. Phys. 17 (1975), p. 19-52 Article
[28] A. Harten, P. Lax & B. van Leer, “On upstream differencing and Godunov-type schemes for hyperbolic conservation laws”, SIAM Rev. 25 (1983), p. 35-61 Article
[29] A. Kurganov & C.-T. Lin, “On the reduction of numerical dissipation in central-upwind schemes”, Commun. Comput. Phys. 2 (2007), p. 141-163
[30] A. Kurganov, S. Noelle & G. Petrova, “Semi-discrete central-upwind scheme for hyperbolic conservation laws and Hamilton-Jacobi equations”, SIAM J. Sci. Comput. 23 (2001), p. 707-740 Article
[31] A. Kurganov & G. Petrova, “A second-order well-balanced positivity preserving central-upwind scheme for the Saint-Venant system”, Commun. Math. Sci. 5 (2007), p. 133-160 Article
[32] A. Kurganov & E. Tadmor, “New high resolution central schemes for nonlinear conservation laws and convection-diffusion equations”, J. Comput. Phys. 160 (2000), p. 241-282 Article
[33] A. Kurganov & E. Tadmor, “Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers”, Numer. Methods Partial Differential Equations 18 (2002), p. 584-608 Article
[34] M.-C. Lai & C. S. Peskin, “An immersed boundary method with formal second-order accuracy and reduced numerical viscosity”, J. Comput. Phys. 160 (2000), p. 705-719 Article
[35] R. J. LeVeque, “The dynamics of pressureless dust clouds and delta waves”, J. Hyperbolic Differ. Equ. 1 (2004), p. 315-327 Article
[36] R. J. LeVeque & D. S. Bale, Wave propagation methods for conservation laws with source terms, Hyperbolic problems: theory, numerics, applications, Vol. II (Zürich, 1998), Internat. Ser. Numer. Math. 130, Birkhäuser, 1999, p. 609–618
[37] G. Li & Y. Xing, “High order finite volume WENO schemes for the Euler equations under gravitational fields”, J. Comput. Phys. 316 (2016), p. 145-163 Article
[38] K.-A. Lie & S. Noelle, “On the artificial compression method for second-order nonoscillatory central difference schemes for systems of conservation laws”, SIAM J. Sci. Comput. 24 (2003) no. 4, p. 1157-1174 Article
[39] J. Luo, K. Xu & N. Liu, “A well-balanced symplecticity-preserving gas-kinetic scheme for hydrodynamic equations under gravitational field”, SIAM J. Sci. Comput. 33 (2011) no. 5, p. 2356-2381 Article
[40] H. Miura & I. I. Glass, “On a dusty-gas shock tube”, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 382 (1982) no. 1783, p. 373-388 Article
[41] A. Neri, T. E. Ongaro, G. Macedonio & D Gidaspow, “Multiparticle simulation of collapsing volcanic columns and pyroclastic flow”, J. Geophys. Res 108 (2003), p. 1-22 Article
[42] H. Nessyahu & E. Tadmor, “Nonoscillatory central differencing for hyperbolic conservation laws”, J. Comput. Phys. 87 (1990) no. 2, p. 408-463 Article
[43] B. Nilsson, O. S. Rozanova & V. M. Shelkovich, “Mass, momentum and energy conservation laws in zero-pressure gas dynamics and $\delta $-shocks: II”, Appl. Anal. 90 (2011) no. 5, p. 831-842 Article
[44] B. Nilsson & V. M. Shelkovich, “Mass, momentum and energy conservation laws in zero-pressure gas dynamics and delta-shocks”, Appl. Anal. 90 (2011) no. 11, p. 1677-1689 Article
[45] M. Pelanti & R. J. Leveque, “High-resolution finite volume methods for dusty gas jets and plumes”, SIAM J. Sci. Comput. 28 (2006), p. 1335-1360 Article
[46] C. S. Peskin, “The immersed boundary method”, Acta Numer. 11 (2002), p. 479-517 Article
[47] P.-A. Raviart, An analysis of particle methods, Numerical methods in fluid dynamics (Como, 1983), Lecture Notes in Math. 1127, Springer, Berlin, 1985, p. 243–324
[48] Yu. G. Rykov, “Propagation of singularities of shock wave type in a system of equations of two-dimensional pressureless gas dynamics”, Mat. Zametki 66 (1999), p. 760-769 (Russian); translation in Math. Notes 66 (1999), pp. 628–635 (2000)
[49] Yu. G. Rykov, “On the nonhamiltonian character of shocks in 2-D pressureless gas”, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 5 (2002), p. 55-78  MR 1881444
[50] T. Saito, “Numerical analysis of dusty-gas flows”, J. Comput. Phys. 176 (2002), p. 129-144 Article
[51] B. Shotorban, G. B. Jacobs, O. Ortiz & Q. Truong, “An Eulerian model for particles nonisothermally carried by a compressible fluid”, Int. J. Heat Mass Transfer 65 (2013), p. 845-854 Article
[52] C.-W. Shu & S. Osher, “Efficient implementation of essentially non-oscillatory shock-capturing schemes”, J. Comput. Phys. 77 (1988), p. 439-471 Article
[53] G. Strang, “On the construction and comparison of difference schemes”, SIAM J. Numer. Anal. 5 (1968), p. 506-517 Article
[54] P. K. Sweby, “High resolution schemes using flux limiters for hyperbolic conservation laws”, SIAM J. Numer. Anal. 21 (1984) no. 5, p. 995-1011 Article
[55] C. T. Tian, K. Xu, K. L. Chan & L. C. Deng, “A three-dimensional multidimensional gas-kinetic scheme for the Navier-Stokes equations under gravitational fields”, J. Comput. Phys. 226 (2007) no. 2, p. 2003-2027 Article
[56] R. Touma, U. Koley & C. Klingenberg, “Well-balanced unstaggered central schemes for the Euler equations with gravitation”, SIAM J. Sci. Comput. 38 (2016) no. 5, p. B773-B807 Article
[57] G. Valentine & K. Wohletz, “Numerical models of Plinian eruption columns and pyroclastic”, J. Geophys. Res 94 (1989), p. 1867-1887 Article
[58] B. van Leer, “Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method”, J. Comput. Phys. 32 (1979) no. 1, p. 101-136 Article
[59] K. H. Wohletz, T. R. McGetchin, M. T. Sandford II & E. M. Jones, “Hydrodynamic aspects of caldera-forming eruptions: numerical models”, J. Geophys. Res 89 (1984), p. 8269-8285 Article
[60] Y. Xing & C.-W. Shu, “High order well-balanced WENO scheme for the gas dynamics equations under gravitational fields”, J. Sci. Comput. 54 (2013) no. 2-3, p. 645-662 Article
[61] K. Xu, J. Luo & S. Chen, “A well-balanced kinetic scheme for gas dynamic equations under gravitational field”, Adv. Appl. Math. Mech. 2 (2010), p. 200-210
[62] M. Yuen, “Some exact blowup solutions to the pressureless Euler equations in $\mathbb{R}^N$”, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) no. 8, p. 2993-2998 Article