Hybrid Finite-Volume-Particle Method for Dusty Gas Flows

Chertock, Alina; Cui, Shumo; Kurganov, Alexander

The SMAI Journal of computational mathematics, Tome 3 (2017), pp. 139-180.

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.

Classification : 65M08, 76M12, 76M28, 86-08, 76M25, 35L65
Mots clés : 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

[1] Botta, N.; Klein, R.; Langenberg, S.; Lützenkirchen, S. Well balanced finite volume methods for nearly hydrostatic flows, J. Comput. Phys., Volume 196 (2004) no. 2, pp. 539-565 | DOI | MR | Zbl

[2] Bouchut, F. On zero pressure gas dynamics, Advances in kinetic theory and computing (Ser. Adv. Math. Appl. Sci.), Volume 22, World Sci. Publ., River Edge, NJ, 1994, pp. 171-190 | MR | Zbl

[3] Bouchut, F.; James, F. Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness, Comm. Partial Differential Equations, Volume 24 (1999) no. 11-12, pp. 2173-2189 | DOI | MR | Zbl

[4] Bouchut, F.; Jin, S.; Li, X. Numerical approximations of pressureless and isothermal gas dynamics, SIAM J. Numer. Anal., Volume 41 (2003), pp. 135-158 | DOI | MR | Zbl

[5] Brenier, Y.; Grenier, E. Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., Volume 35 (1998), pp. 2317-2328 | DOI | MR | Zbl

[6] Carcano, S.; Bonaventura, L.; Esposti Ongaro, T.; Neri, A. A semi-implicit, second order accurate numerical model for multiphase underexpanded volcanic jets, Geosci. Model Dev. Discuss., Volume 6 (2013) no. 1, pp. 399-452 | DOI

[7] Chandrashekar, P.; Klingenberg, C. A second order well-balanced finite volume scheme for Euler equations with gravity, SIAM J. Sci. Comput., Volume 37 (2015) no. 3, p. B382-B402 | DOI | MR | Zbl

[8] Chen, G.-Q.; Liu, H. 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., Volume 34 (2003), pp. 925-938 | DOI | MR | Zbl

[9] Chertock, A.; Cui, S.; Kurganov, A.; Özcan, Ş. N.; Tadmor, E. Well-balanced central-upwind schemes for the Euler equations with gravitation (Submitted)

[10] Chertock, A.; Cui, S.; Kurganov, A.; Wu, T. Well-balanced positivity preserving central-upwind scheme for the shallow water system with friction terms, Internat. J. Numer. Meth. Fluids, Volume 78 (2015), pp. 355-383 | DOI | MR

[11] Chertock, A.; Kurganov, A. On a hybrid finite-volume particle method, M2AN Math. Model. Numer. Anal, Volume 38 (2004), pp. 1071-1091 | DOI | Numdam | MR | Zbl

[12] Chertock, A.; Kurganov, A. On a practical implementation of particle methods, Appl. Numer. Math., Volume 56 (2006), pp. 1418-1431 | DOI | MR | Zbl

[13] Chertock, A.; Kurganov, A.; Petrova, G. Finite-volume-particle methods for models of transport of pollutant in shallow water, J. Sci. Comput., Volume 27 (2006), pp. 189-199 | DOI | MR | Zbl

[14] Chertock, A.; Kurganov, A.; Rykov, Yu. A new sticky particle method for pressureless gas dynamics, SIAM J. Numer. Anal., Volume 45 (2007), pp. 2408-2441 | DOI | MR | Zbl

[15] Cottet, G.-H.; Koumoutsakos, P. D. Vortex methods, Cambridge University Press, Cambridge, 2000

[16] Dartevelle, S.; Rose, W.; Stix, J.; Kelfoun, K.; Vallance, J.W. Numerical modeling of geophysical granular flows: 2. Computer simulations of plinian clouds and pyroclastic flows and surges, Geochem. Geophys. Geosyst., Volume 5 (2004) no. 8 | DOI

[17] Desveaux, V.; Zenk, M.; Berthon, C.; Klingenberg, C. 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.), Volume 77, Springer, Cham, 2014, pp. 217-226 | MR | Zbl

[18] Dobran, F.; Neri, A.; Macedonio, G. Numerical simulation of collapsing volcanic columns, J. Geophys. Res., Volume 98 (1993), pp. 4231-4259 | DOI

[19] Dufek, J.; Bergantz, G. W. Dynamics and deposits generated by the Kos Plateau Tuff eruption: Controls of basal particle loss on pyroclastic flow transport, Geochem. Geophys. Geosyst., Volume 8 (2007) no. 12 | DOI

[20] E, W.; Rykov, Yu. G.; Sinai, Ya. G. 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., Volume 177 (1996), pp. 349-380 | DOI | MR | Zbl

[21] Einfeld, B. On Godunov-type methods for gas dynamics, SIAM J. Numer. Anal., Volume 25 (1988), pp. 294-318 | DOI | MR

[22] Glaister, P. Flux difference splitting for the Euler equations with axial symmetry, J. Engrg. Math., Volume 22 (1988) no. 2, pp. 107-121 | DOI | MR | Zbl

[23] Gottlieb, S.; Ketcheson, D.; Shu, C.-W. Strong stability preserving Runge-Kutta and multistep time discretizations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011, xii+176 pages | Zbl

[24] Gottlieb, S.; Shu, C.-W.; Tadmor, E. Strong stability-preserving high-order time discretization methods, SIAM Rev., Volume 43 (2001), pp. 89-112 | DOI | MR | Zbl

[25] Gurris, M.; Kuzmin, D.; Turek, S. Finite element simulation of compressible particle-laden gas flows, J. Comput. Appl. Math., Volume 233 (2010) no. 12, pp. 3121-3129 | DOI | MR | Zbl

[26] Hank, S.; Saurel, R.; Le Metayer, O. A hyperbolic Eulerian model for dilute two-phase suspensions, Journal of Modern Physics, Volume 2 (2011), pp. 997-1011 | DOI

[27] Harlow, A.; Amsden, A. A. Numerical calculation of multiphase fluid flow, J. Comput. Phys., Volume 17 (1975), pp. 19-52 | DOI | Zbl

[28] Harten, A.; Lax, P.; van Leer, B. On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., Volume 25 (1983), pp. 35-61 | DOI | MR | Zbl

[29] Kurganov, A.; Lin, C.-T. On the reduction of numerical dissipation in central-upwind schemes, Commun. Comput. Phys., Volume 2 (2007), pp. 141-163 | MR | Zbl

[30] Kurganov, A.; Noelle, S.; Petrova, G. Semi-discrete central-upwind scheme for hyperbolic conservation laws and Hamilton-Jacobi equations, SIAM J. Sci. Comput., Volume 23 (2001), pp. 707-740 | DOI

[31] Kurganov, A.; Petrova, G. A second-order well-balanced positivity preserving central-upwind scheme for the Saint-Venant system, Commun. Math. Sci., Volume 5 (2007), pp. 133-160 | DOI | MR | Zbl

[32] Kurganov, A.; Tadmor, E. New high resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys., Volume 160 (2000), pp. 241-282 | DOI | MR | Zbl

[33] Kurganov, A.; Tadmor, E. Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers, Numer. Methods Partial Differential Equations, Volume 18 (2002), pp. 584-608 | DOI | MR | Zbl

[34] Lai, M.-C.; Peskin, C. S. An immersed boundary method with formal second-order accuracy and reduced numerical viscosity, J. Comput. Phys., Volume 160 (2000), pp. 705-719 | DOI | MR | Zbl

[35] LeVeque, R. J. The dynamics of pressureless dust clouds and delta waves, J. Hyperbolic Differ. Equ., Volume 1 (2004), pp. 315-327 | DOI | MR | Zbl

[36] LeVeque, R. J.; Bale, D. S. Wave propagation methods for conservation laws with source terms, Hyperbolic problems: theory, numerics, applications, Vol. II (Zürich, 1998) (Internat. Ser. Numer. Math.), Volume 130, Birkhäuser, Basel, 1999, pp. 609-618 | MR | Zbl

[37] Li, G.; Xing, Y. High order finite volume WENO schemes for the Euler equations under gravitational fields, J. Comput. Phys., Volume 316 (2016), pp. 145-163 | DOI | MR | Zbl

[38] Lie, K.-A.; Noelle, S. On the artificial compression method for second-order nonoscillatory central difference schemes for systems of conservation laws, SIAM J. Sci. Comput., Volume 24 (2003) no. 4, pp. 1157-1174 | DOI | MR | Zbl

[39] Luo, J.; Xu, K.; Liu, N. A well-balanced symplecticity-preserving gas-kinetic scheme for hydrodynamic equations under gravitational field, SIAM J. Sci. Comput., Volume 33 (2011) no. 5, pp. 2356-2381 | DOI | MR | Zbl

[40] Miura, H.; Glass, I. I. On a dusty-gas shock tube, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., Volume 382 (1982) no. 1783, pp. 373-388 | DOI

[41] Neri, A.; Ongaro, T. E.; Macedonio, G.; Gidaspow, D Multiparticle simulation of collapsing volcanic columns and pyroclastic flow, J. Geophys. Res, Volume 108 (2003), pp. 1-22 | DOI

[42] Nessyahu, H.; Tadmor, E. Nonoscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys., Volume 87 (1990) no. 2, pp. 408-463 | DOI | MR | Zbl

[43] Nilsson, B.; Rozanova, O. S.; Shelkovich, V. M. Mass, momentum and energy conservation laws in zero-pressure gas dynamics and $\delta$-shocks: II, Appl. Anal., Volume 90 (2011) no. 5, pp. 831-842 | DOI | MR | Zbl

[44] Nilsson, B.; Shelkovich, V. M. Mass, momentum and energy conservation laws in zero-pressure gas dynamics and delta-shocks, Appl. Anal., Volume 90 (2011) no. 11, pp. 1677-1689 | DOI | MR | Zbl

[45] Pelanti, M.; Leveque, R. J. High-resolution finite volume methods for dusty gas jets and plumes, SIAM J. Sci. Comput., Volume 28 (2006), pp. 1335-1360 | DOI | MR | Zbl

[46] Peskin, C. S. The immersed boundary method, Acta Numer., Volume 11 (2002), pp. 479-517 | DOI | MR | Zbl

[47] Raviart, P.-A. An analysis of particle methods, Numerical methods in fluid dynamics (Como, 1983) (Lecture Notes in Math.), Volume 1127, Springer, Berlin, 1985, pp. 243-324 | MR

[48] Rykov, Yu. G. Propagation of singularities of shock wave type in a system of equations of two-dimensional pressureless gas dynamics, Mat. Zametki, Volume 66 (1999), p. 760-769 (Russian); translation in Math. Notes 66 (1999), pp. 628–635 (2000) | MR | Zbl

[49] Rykov, Yu. G. On the nonhamiltonian character of shocks in 2-D pressureless gas, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., Volume (8) 5 (2002), pp. 55-78 | MR | Zbl

[50] Saito, T. Numerical analysis of dusty-gas flows, J. Comput. Phys., Volume 176 (2002), pp. 129-144 | DOI | Zbl

[51] Shotorban, B.; Jacobs, G. B.; Ortiz, O.; Truong, Q. An Eulerian model for particles nonisothermally carried by a compressible fluid, Int. J. Heat Mass Transfer, Volume 65 (2013), pp. 845-854 | DOI

[52] Shu, C.-W.; Osher, S. Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., Volume 77 (1988), pp. 439-471 | DOI

[53] Strang, G. On the construction and comparison of difference schemes, SIAM J. Numer. Anal., Volume 5 (1968), pp. 506-517 | DOI | MR | Zbl

[54] Sweby, P. K. High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Numer. Anal., Volume 21 (1984) no. 5, pp. 995-1011 | DOI | MR | Zbl

[55] Tian, C. T.; Xu, K.; Chan, K. L.; Deng, L. C. A three-dimensional multidimensional gas-kinetic scheme for the Navier-Stokes equations under gravitational fields, J. Comput. Phys., Volume 226 (2007) no. 2, pp. 2003-2027 | DOI | MR | Zbl

[56] Touma, R.; Koley, U.; Klingenberg, C. Well-balanced unstaggered central schemes for the Euler equations with gravitation, SIAM J. Sci. Comput., Volume 38 (2016) no. 5, p. B773-B807 | DOI | MR | Zbl

[57] Valentine, G.; Wohletz, K. Numerical models of Plinian eruption columns and pyroclastic, J. Geophys. Res, Volume 94 (1989), pp. 1867-1887 | DOI

[58] van Leer, B. Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method, J. Comput. Phys., Volume 32 (1979) no. 1, pp. 101-136 | DOI | Zbl

[59] Wohletz, K. H.; McGetchin, T. R.; Sandford II, M. T.; Jones, E. M. Hydrodynamic aspects of caldera-forming eruptions: numerical models, J. Geophys. Res, Volume 89 (1984), pp. 8269-8285 | DOI

[60] Xing, Y.; Shu, C.-W. High order well-balanced WENO scheme for the gas dynamics equations under gravitational fields, J. Sci. Comput., Volume 54 (2013) no. 2-3, pp. 645-662 | DOI | MR | Zbl

[61] Xu, K.; Luo, J.; Chen, S. A well-balanced kinetic scheme for gas dynamic equations under gravitational field, Adv. Appl. Math. Mech., Volume 2 (2010), pp. 200-210 | MR

[62] Yuen, M. Some exact blowup solutions to the pressureless Euler equations in ${ℝ}^N$, Commun. Nonlinear Sci. Numer. Simul., Volume 16 (2011) no. 8, pp. 2993-2998 | DOI | MR | Zbl