Uniform regularity in the random space and spectral accuracy of the stochastic Galerkin method for a kinetic-fluid two-phase flow model with random initial inputs in the light particle regime

Shu, Ruiwen; Jin, Shi

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 5, pp. 1651-1678.

We consider a kinetic-fluid model with random initial inputs which describes disperse two-phase flows. In the light particle regime, using energy estimates, we prove the uniform regularity in the random space of the model for random initial data near the global equilibrium in some suitable Sobolev spaces, with the randomness in the initial particle distribution and fluid velocity. By hypocoercivity arguments, we prove that the energy decays exponentially in time, which means that the long time behavior of the solution is insensitive to such randomness in the initial data. Then we consider the generalized polynomial chaos stochastic Galerkin method (gPC-sG) for the same model. For initial data near the global equilibrium and smooth enough in the physical and random spaces, we prove that the gPC-sG method has spectral accuracy, uniformly in time and the Knudsen number, and the error decays exponentially in time.

Classification : 35Q35, 65L60
Mots clés : Two-phase flow, kinetic theory, uncertainty quantification, stochastic Galerkin method, hypocoercivity

[1] M.J. Andrews and P.J. O’Rourke, The multiphase particle-in-cell (mp-pic) method for dense particulate flows. Int. J. Multiph. Flow 22 (1996) 379–402. | DOI | Zbl

[2] I. Babuska, R. Tempone and G.E. Zouraris, Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42 (2004) 800–825. | DOI | MR | Zbl

[3] I. Babuska, F. Nobile and R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45 (2007) 1005–1034. | DOI | MR | Zbl

[4] J. Back, F. Nobile, L. Tamellini and R. Tempone, Stochastic spectral galerkin and collocation methods for pdes with random coefficients: a numerical comparison, in Spectral and High Order Methods for Partial Differential Equations, edited by E.M. Rønquist and J.S. Hesthaven. Springer-Verlag, Berlin, Heidelberg (2011). | DOI | MR | Zbl

[5] R. Caflisch and G. Papanicolaou, Dynamic theory of suspensions with Brownian effects. SIAM J. Appl. Math. 43 (1983). | DOI | MR | Zbl

[6] R.G. Ghanem and P.D. Spanos, Stochastic Finite Elements: A Spectral Approach. Springer-Verlag, New York (1991). | DOI | MR | Zbl

[7] D. Gidaspow, R. Bezburuah and J. Ding, Hydrodynamics of Circulating Fluidized Beds: Kinetic Theory Approach. Illinois Institute of Technology, Department of Chemical Engineering, Chicago, IL, USA (1991).

[8] A.D. Gosman and E. Loannides, Aspects of computer simulation of liquid-fueled combustors. J. Energy 7 (1983) 482–490. | DOI

[9] T. Goudon, P.-E. Jabin and A. Vasseur, Hydrodynamic limit for the Vlasov-Navier-Stokes equations I. Light particles regime. Indiana Univ. Math. J. 53 (2004) 1495–1516. | DOI | MR | Zbl

[10] T. Goudon, P.-E. Jabin and A. Vasseur, Hydrodynamic limit for the Vlasov-Navier-Stokes equations II. Fine particles regime. Indiana Univ. Math. J. 53 (2004) 1517–1536. | DOI | MR | Zbl

[11] T. Goudon, L. He, A. Moussa and P. Zhang, The Navier-Stokes-Vlasov-Fokker-Planck system near equilibrium. SIAM J. Math. Anal. 42 (2010) 2177–2202. | DOI | MR | Zbl

[12] T. Goudon, S. Jin, J.-G. Liu and B. Yan, Asymptotic-preserving schemes for kinetic-fluid modeling of disperse two-phase flows. J. Comput. Phys. 246 (2013) 145–164 | DOI | MR | Zbl

[13] M.D. Gunzburger, C.G. Webster and G. Zhang, Stochastic finite element methods for partial differential equations with random input data. Acta Numer. 23 (2014) 521–650. | DOI | MR | Zbl

[14] P.-E. Jabin and B. Perthame, Notes on mathematical problems on the dynamics of dispersed particles interacting through a fluid, in Modeling in Applied Sciences, A Kinetic Theory Approach, edited by N. Bellomo and M. Pulvirenti. Birkhauser (2000) 111–147. | DOI | MR | Zbl

[15] X. Jiang, G.A. Siamas, K. Jagus and T.G. Karayiannis, Physical modelling and advanced simulations of gas–liquid two-phase jet flows in atomization and sprays. Prog. Energy Combust. Sci. 36 (2010) 131–167. | DOI

[16] S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput. 21 (1999) 441–454. | DOI | MR | Zbl

[17] S. Jin and L. Liu, An asymptotic-preserving stochastic Galerkin method for the semiconductor Boltzmann equation with random inputs and diffusive scalings. SIAM Multiscale Model. Simul. 15 (2017) 157–183. | DOI | MR | Zbl

[18] S. Jin and R. Shu, A stochastic asymptotic-preserving scheme for a kinetic-fluid model for disperse two-phase flows with uncertainty. J. Comput. Phys. 335 (2017) 905–924. | DOI | MR | Zbl

[19] S. Jin and Y. Zhu, Hypocoercivity and uniform regularity for the Vlasov-Poisson-Fokker-Planck system with uncertainty and multiple scales. SIAM J. Math. Anal. 50 (2018) 1790–1816. | DOI | MR | Zbl

[20] S. Jin, D. Xiu and X. Zhu, Asymptotic-preserving methods for hyperbolic and transport equations with random inputs and diffusive scalings. J. Comput. Phys. 289 (2015) 35–52. | DOI | MR | Zbl

[21] S. Jin, J.-G. Liu and Z. Ma, Uniform spectral convergence of the stochastic Galerkin method for the linear transport equations with random inputs in diffusive regime and a micro-macro decomposition based asymptotic preserving method. Res. Math. Sci. 4 (2017) 15. | DOI | MR | Zbl

[22] Q. Li and L. Wang, Uniform regularity for linear kinetic equations with random input based on hypocoercivity. SIAM Uncertain. Quantif. 5 (2017) 1193–1219. | DOI | MR | Zbl

[23] L. Liu, Uniform spectral convergence of the stochastic Galerkin method for the linear semiconductor Boltzmann equation with random inputs and diffusive scalings. Kinet. Relat. Model. 11 (2018) 1139–1156. | DOI | MR | Zbl

[24] O.P. Le Maître and O.M. Knio, Spectral Methods for Uncertainty Quantification, Scientific Computation, With Applications to Computational Fluid Dynamics. Springer, New York (2010). | MR | Zbl

[25] H. Niederreiter, P. Hellekalek, G. Larcher and P. Zinterhof, Monte Carlo and Quasi-Monte Carlo Methods 1996. Springer-Verlag (1998). | DOI | MR | Zbl

[26] F. Nobile, R. Tempone and C. G. Webster, A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 46 (2008) 2309–2345. | DOI | MR | Zbl

[27] P.J. O’Rourke, Collective drop effects on vaporizing liquid sprays. Los Alamos National Lab., NM (USA) (1981).

[28] G. Szegő, Orthogonal Polynomials. American Mathematical Society (1939). | MR

[29] F.A. Williams, Combustion Theory, 2nd edition. Benjamin Cummings Publ. (1985).

[30] D. Xiu, Fast numerical methods for stochastic computations: a review. Commun. Comput. Phys. 5 (2009) 242–272. | MR | Zbl

[31] D. Xiu, Numerical Methods for Stochastic Computation. Princeton University Press, Princeton, New Jersey (2010). | Zbl

[32] D. Xiu and J.S. Hesthaven, High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27 (2005) 1118–1139. | DOI | MR | Zbl

[33] D. Xiu and G.E. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24 (2002) 619–644. | DOI | MR | Zbl