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August  2019, 12(4): 909-922. doi: 10.3934/krm.2019034

Spectral convergence of the stochastic galerkin approximation to the boltzmann equation with multiple scales and large random perturbation in the collision kernel

1. 

Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstrasse 8–10, 1040 Wien, Austria

2. 

School of Mathematical Sciences, Institute of Natural Sciences, MOE-LSC and SHL-MAC, Shanghai Jiao Tong University, Shanghai 200240, China

3. 

Institute for Computational Engineering and Sciences, University of Texas at Austin, Austin, Texas 78705, USA

Received  October 2018 Revised  January 2019 Published  May 2019

Fund Project: The first author acknowledges partial support from the Austrian Science Fund (FWF), grants P27352 and P30000, the second author is supported by NSF grants DMS-1522184, DMS-1819012 and DMS-1107291: RNMS KI-Net, NSFC grants No. 31571071 and No. 11871297, the third author is supported by the funding DOE–Simulation Center for Runaway Electron Avoidance and Mitigation, project No. DE-SC0016283.

In [L. Liu and S. Jin, Multiscale Model. Simult., 16, 1085-1114, 2018], spectral convergence and long-time decay of the numerical solution towards the global equilibrium of the stochastic Galerkin approximation for the Boltzmann equation with random inputs in the initial data and collision kernel for hard potentials and Maxwellian molecules under Grad's angular cutoff were established using the hypocoercive properties of the collisional kinetic model. One assumption for the random perturbation of the collision kernel is that the perturbation is in the order of the Knudsen number, which can be very small in the fluid dynamical regime. In this article, we remove this smallness assumption, and establish the same results but now for random perturbations of the collision kernel that can be of order one. The new analysis relies on the establishment of a spectral gap for the numerical collision operator.

Citation: Esther S. Daus, Shi Jin, Liu Liu. Spectral convergence of the stochastic galerkin approximation to the boltzmann equation with multiple scales and large random perturbation in the collision kernel. Kinetic & Related Models, 2019, 12 (4) : 909-922. doi: 10.3934/krm.2019034
References:
[1]

F. Bouchut, F. Golse and M. Pulvirenti, Kinetic Equations and Asymptotic Theory, Elsevier, 2000.  Google Scholar

[2]

M. Briant, From the Boltzmann equation to the incompressible Navier-Stokes equations on the torus: A quantitative error estimate, Journal of Differential Equations, 259 (2015), 6072-6141.  doi: 10.1016/j.jde.2015.07.022.  Google Scholar

[3]

C. Cercignani, The Boltzmann equation in the whole space, in The Boltzmann Equation and Its Applications, Springer, 1988, 40–103. doi: 10.1007/978-1-4612-1039-9.  Google Scholar

[4]

E. S. DausA. JüngelC. Mouhot and N. Zamponi, Hypocoercivity for a linearized multispecies Boltzmann system, SIAM J. Math. Anal., 48 (2016), 538-568.  doi: 10.1137/15M1017934.  Google Scholar

[5]

L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation, Inventiones mathematicae, 159 (2005), 245-316.  doi: 10.1007/s00222-004-0389-9.  Google Scholar

[6]

R. G. Ghanem and P. D. Spanos, Stochastic Finite Elements: A Spectral Approach, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-3094-6.  Google Scholar

[7]

Y. Guo, Boltzmann diffusive limit beyond the Navier-Stokes approximation, Communications on Pure and Applied Mathematics, 59 (2006), 626-687.  doi: 10.1002/cpa.20121.  Google Scholar

[8]

F. Hérau and F. Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Archive for Rational Mechanics and Analysis, 171 (2004), 151-218.  doi: 10.1007/s00205-003-0276-3.  Google Scholar

[9]

J. Hu and S. Jin, A stochastic Galerkin method for the Boltzmann equation with uncertainty, J. Comput. Phys., 315 (2016), 150-168.  doi: 10.1016/j.jcp.2016.03.047.  Google Scholar

[10]

______, Uncertainty quantification for kinetic equations, in Uncertainty Quantification for Hyperbolic and Kinetic Equations, vol. 14 of SEMA SIMAI Springer Ser., Springer, Cham, 2017,193–229. doi: 10.1007/978-3-319-67110-9_6.  Google Scholar

[11]

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), 25pp. doi: 10.1186/s40687-017-0105-1.  Google Scholar

[12]

Q. Li and L. Wang, Uniform regularity for linear kinetic equations with random input based on hypocoercivity, SIAM/ASA J. Uncertain. Quantif., 5 (2017), 1193-1219.  doi: 10.1137/16M1106675.  Google Scholar

[13]

L. Liu, Uniform spectral convergence of the stochastic Galerkin method for the linear semiconductor Boltzmann equation with random inputs and diffusive scaling, Kinet. Relat. Models, 11 (2018), 1139-1156.  doi: 10.3934/krm.2018044.  Google Scholar

[14]

______, A stochastic asymptotic-preserving scheme for the bipolar semiconductor BoltzmannPoisson system with random inputs and diffusive scalings, J. Comput. Phys., 376 (2019), 634–659. doi: 10.1016/j.jcp.2018.09.055.  Google Scholar

[15]

L. Liu and S. Jin, Hypocoercivity based sensitivity analysis and spectral convergence of the stochastic Galerkin approximation to collisional kinetic equations with multiple scales and random inputs, Multiscale Model. Simul., 16 (2018), 1085-1114.  doi: 10.1137/17M1151730.  Google Scholar

[16]

C. Mouhot, Explicit coercivity estimates for the linearized Boltzmann and Landau operators, Comm. Partial Differential Equations, 31 (2006), 1321-1348.  doi: 10.1080/03605300600635004.  Google Scholar

[17]

C. Mouhot and L. Neumann, Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity, 19 (2006), 969-998.  doi: 10.1088/0951-7715/19/4/011.  Google Scholar

[18]

R. Shu and S. Jin, 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, ESAIM Math. Model. Numer. Anal., 52 (2018), 1651-1678.  doi: 10.1051/m2an/2018024.  Google Scholar

[19]

R. M. Strain and Y. Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations, 31 (2006), 417-429.  doi: 10.1080/03605300500361545.  Google Scholar

[20]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), iv+141 pp. doi: 10.1090/S0065-9266-09-00567-5.  Google Scholar

[21] D. Xiu, Numerical Methods for Stochastic Computations, Princeton University Press, Princeton, New Jersey, 2010.   Google Scholar
[22]

D. Xiu and J. Shen, Efficient stochastic Galerkin methods for random diffusion equations, J. Comput. Phys., 228 (2009), 266-281.  doi: 10.1016/j.jcp.2008.09.008.  Google Scholar

show all references

References:
[1]

F. Bouchut, F. Golse and M. Pulvirenti, Kinetic Equations and Asymptotic Theory, Elsevier, 2000.  Google Scholar

[2]

M. Briant, From the Boltzmann equation to the incompressible Navier-Stokes equations on the torus: A quantitative error estimate, Journal of Differential Equations, 259 (2015), 6072-6141.  doi: 10.1016/j.jde.2015.07.022.  Google Scholar

[3]

C. Cercignani, The Boltzmann equation in the whole space, in The Boltzmann Equation and Its Applications, Springer, 1988, 40–103. doi: 10.1007/978-1-4612-1039-9.  Google Scholar

[4]

E. S. DausA. JüngelC. Mouhot and N. Zamponi, Hypocoercivity for a linearized multispecies Boltzmann system, SIAM J. Math. Anal., 48 (2016), 538-568.  doi: 10.1137/15M1017934.  Google Scholar

[5]

L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation, Inventiones mathematicae, 159 (2005), 245-316.  doi: 10.1007/s00222-004-0389-9.  Google Scholar

[6]

R. G. Ghanem and P. D. Spanos, Stochastic Finite Elements: A Spectral Approach, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-3094-6.  Google Scholar

[7]

Y. Guo, Boltzmann diffusive limit beyond the Navier-Stokes approximation, Communications on Pure and Applied Mathematics, 59 (2006), 626-687.  doi: 10.1002/cpa.20121.  Google Scholar

[8]

F. Hérau and F. Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Archive for Rational Mechanics and Analysis, 171 (2004), 151-218.  doi: 10.1007/s00205-003-0276-3.  Google Scholar

[9]

J. Hu and S. Jin, A stochastic Galerkin method for the Boltzmann equation with uncertainty, J. Comput. Phys., 315 (2016), 150-168.  doi: 10.1016/j.jcp.2016.03.047.  Google Scholar

[10]

______, Uncertainty quantification for kinetic equations, in Uncertainty Quantification for Hyperbolic and Kinetic Equations, vol. 14 of SEMA SIMAI Springer Ser., Springer, Cham, 2017,193–229. doi: 10.1007/978-3-319-67110-9_6.  Google Scholar

[11]

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), 25pp. doi: 10.1186/s40687-017-0105-1.  Google Scholar

[12]

Q. Li and L. Wang, Uniform regularity for linear kinetic equations with random input based on hypocoercivity, SIAM/ASA J. Uncertain. Quantif., 5 (2017), 1193-1219.  doi: 10.1137/16M1106675.  Google Scholar

[13]

L. Liu, Uniform spectral convergence of the stochastic Galerkin method for the linear semiconductor Boltzmann equation with random inputs and diffusive scaling, Kinet. Relat. Models, 11 (2018), 1139-1156.  doi: 10.3934/krm.2018044.  Google Scholar

[14]

______, A stochastic asymptotic-preserving scheme for the bipolar semiconductor BoltzmannPoisson system with random inputs and diffusive scalings, J. Comput. Phys., 376 (2019), 634–659. doi: 10.1016/j.jcp.2018.09.055.  Google Scholar

[15]

L. Liu and S. Jin, Hypocoercivity based sensitivity analysis and spectral convergence of the stochastic Galerkin approximation to collisional kinetic equations with multiple scales and random inputs, Multiscale Model. Simul., 16 (2018), 1085-1114.  doi: 10.1137/17M1151730.  Google Scholar

[16]

C. Mouhot, Explicit coercivity estimates for the linearized Boltzmann and Landau operators, Comm. Partial Differential Equations, 31 (2006), 1321-1348.  doi: 10.1080/03605300600635004.  Google Scholar

[17]

C. Mouhot and L. Neumann, Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity, 19 (2006), 969-998.  doi: 10.1088/0951-7715/19/4/011.  Google Scholar

[18]

R. Shu and S. Jin, 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, ESAIM Math. Model. Numer. Anal., 52 (2018), 1651-1678.  doi: 10.1051/m2an/2018024.  Google Scholar

[19]

R. M. Strain and Y. Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations, 31 (2006), 417-429.  doi: 10.1080/03605300500361545.  Google Scholar

[20]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), iv+141 pp. doi: 10.1090/S0065-9266-09-00567-5.  Google Scholar

[21] D. Xiu, Numerical Methods for Stochastic Computations, Princeton University Press, Princeton, New Jersey, 2010.   Google Scholar
[22]

D. Xiu and J. Shen, Efficient stochastic Galerkin methods for random diffusion equations, J. Comput. Phys., 228 (2009), 266-281.  doi: 10.1016/j.jcp.2008.09.008.  Google Scholar

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