October  2019, 12(5): 969-993. doi: 10.3934/krm.2019037

Local sensitivity analysis and spectral convergence of the stochastic Galerkin method for discrete-velocity Boltzmann equations with multi-scales and random inputs

1. 

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

2. 

Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, USA

Received  May 2018 Revised  February 2019 Published  July 2019

Fund Project: This work was partially supported by the NSFC grants No. 31571071 and No. 11871297.

In this paper we study the general discrete-velocity models of Boltzmann equation with uncertainties from collision kernel and random inputs. We follow the framework of Kawashima and extend it to the case of diffusive scaling in a random setting. First, we provide a uniform regularity analysis in the random space with the help of a Lyapunov-type functional, and prove a uniformly (in the Knudsen number) exponential decay towards the global equilibrium, under certain smallness assumption on the random perturbation of the collision kernel, for suitably small initial data. Then we consider the generalized polynomial chaos based stochastic Galerkin approximation (gPC-SG) of the model, and prove the spectral convergence and the exponential time decay of the gPC-SG error uniformly in the Knudsen number.

Citation: Shi Jin, Yingda Li. Local sensitivity analysis and spectral convergence of the stochastic Galerkin method for discrete-velocity Boltzmann equations with multi-scales and random inputs. Kinetic and Related Models, 2019, 12 (5) : 969-993. doi: 10.3934/krm.2019037
References:
[1]

A. BarthC. Schwab and N. Zollinger, Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients, Numerische Mathematik, 119 (2011), 123-161.  doi: 10.1007/s00211-011-0377-0.

[2]

A. Bellouquid, A diffusive limit for nonlinear discrete velocity models, Mathematical Models and Methods in Applied Sciences, 13 (2003), 35-58.  doi: 10.1142/S0218202503002374.

[3]

J. E. Broadwell, Shock structure in a simple discrete velocity gas, The Physics of Fluids, 7 (1964), 1243-1247.  doi: 10.1063/1.1711368.

[4]

T. Carleman, Problemes Mathématiques dans la Théorie Cinétique de Gaz, vol. 2, Almqvist & Wiksell, 1957.

[5]

J. CharrierR. Scheichl and A. L. Teckentrup, Finite element error analysis of elliptic PDEs with random coefficients and its application to multilevel Monte Carlo methods, SIAM Journal on Numerical Analysis, 51 (2013), 322-352.  doi: 10.1137/110853054.

[6]

S. Chen and G. D. Doolen, Lattice boltzmann method for fluid flows, Annual Review of Fluid Mechanics, 30 (1998), 329-364.  doi: 10.1146/annurev.fluid.30.1.329.

[7]

C. Villani, Hypocoercivity, American Mathematical Soc., 2009. doi: 10.1090/S0065-9266-09-00567-5.

[8]

F. Golse, S. Jin and C. D. Levermore, The convergence of numerical transfer schemes in diffusive regimes Ⅰ: Discrete-ordinate method, SIAM Journal on Numerical Analysis, 36 (1999), 1333–1369, http://epubs.siam.org/doi/abs/10.1137/S0036142997315986. doi: 10.1137/S0036142997315986.

[9]

S.-Y. Ha and A. E. Tzavaras, Lyapunov functionals and l1-stability for discrete velocity boltzmann equations, Communications in Mathematical Physics, 239 (2003), 65-92.  doi: 10.1007/s00220-003-0866-9.

[10]

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.

[11]

J. Hu and S. Jin, A Stochastic Galerkin method for the Boltzmann equation with uncertainty, Journal of Computational Physics, 315 (2016), 150–168, http://dx.doi.org/10.1016/j.jcp.2016.03.047. doi: 10.1016/j.jcp.2016.03.047.

[12]

J. Hu and S. Jin, Uncertainty quantification for kinetic equations, Uncertainty Quantification for Hyperbolic and Kinetic Equations, 14 (2018), 193-229.  doi: 10.1007/978-3-319-67110-9_6.

[13]

R. Illner and M. C. Reed, The decay of solution of the Carleman model, Mathematical Method in the Applied Sciences, 3 (1981), 121-127.  doi: 10.1002/mma.1670030110.

[14]

R. Illner and M. C. Reed, Decay to equilibrium for the Carleman model in a box, SIAM Journal on Applied Mathematics, 44 (1984), 1067-1075.  doi: 10.1137/0144076.

[15]

K. Inoue and T. Nishida, On the Broadwell model of the Boltzmann equation for a simple discrete velocity gas, Applied Mathematics and Optimization, 3 (1976), 27-49.  doi: 10.1007/BF02106189.

[16]

S. Jin, Efficient asymptotic-preserving (ap) schemes for some multiscale kinetic equations, SIAM Journal on Scientific Computing, 21 (1999), 441-454.  doi: 10.1137/S1064827598334599.

[17]

S. Jin, Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review, Lecture Notes for Summer School on Methods and Models of Kinetic Theory (M & MKT), Porto Ercole (Grosseto, Italy), 3 (2012), 177–216.

[18]

S. Jin, J.-G. J. 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 acro decomposition-based asymptotic-preserving method, Res Math Sci, 4 (2017), Paper No. 15, 25 pp. doi: 10.1186/s40687-017-0105-1.

[19]

S. Jin and L. Liu, An asymptotic-preserving stochastic galerkin method for the semiconductor boltzmann equation with random inputs and diffusive scalings, Siam Journal on Multiscale Model & Simulation, 15 (2017), 157-183.  doi: 10.1137/15M1053463.

[20]

S. JinL. Pareschi and G. Toscani, Diffusive relaxation schemes for multiscale discrete-velocity kinetic equations, SIAM Journal on Numerical Analysis, 35 (1998), 2405-2439.  doi: 10.1137/S0036142997315962.

[21]

S. JinD. Xiu and X. Zhu, Asymptotic-preserving methods for hyperbolic and transport equations with random inputs and diffusive scalings, Journal of Computational Physics, 289 (2015), 35-52.  doi: 10.1016/j.jcp.2015.02.023.

[22]

S. Jin and Y. Zhu, Hypocoercivity and uniform regularity for the vlasov–poisson–fokker–planck system with uncertainty and multiple scales, SIAM Journal on Mathematical Analysis, 50 (2018), 1790-1816.  doi: 10.1137/17M1123845.

[23]

S. Kawashima, Global existence and stability of solutions for discrete velocity models of the Boltzmann equation, North-Holland Mathematics Studies, 98 (1984), 59–85, https://www.sciencedirect.com/science/article/pii/S0304020808714920. doi: 10.1016/S0304-0208(08)71492-0.

[24]

S. Kawashima, Smooth global solutions for two-dimensional equations of electro-magneto-fluid dynamics, Japan Journal of Applied Mathematics, 1 (1984), 207-222.  doi: 10.1007/BF03167869.

[25]

S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, PhD thesis, Kyoto University, 1984.

[26]

S. Kawashima, Large-time Behavior of Solutions of the Discrete Boltzmann Equation, Physics, 589 (1987), 563-589.  doi: 10.1007/BF01208958.

[27]

S. Kawashima, The boltzmann equation and thirteen moments, Japan Journal of Applied Mathematics, 7 (1990), 301-320.  doi: 10.1007/BF03167846.

[28]

S. KawashimaA. Matsumura and T. Nishida, On the fluid-dynamical approximation to the Boltzmann equation at the level of the Navier-Stokes equation, Communications in Mathematical Physics, 70 (1979), 97-124.  doi: 10.1007/BF01982349.

[29]

S. KawashimaM. Okada and Ot hers, Smooth global solutions for the one-dimensional equations in magnetohydrodynamics, Proceedings of the Japan Academy, Series A, Mathematical Sciences, 58 (1982), 384-387.  doi: 10.3792/pjaa.58.384.

[30]

A. Klar, An asymptotic-induced scheme for nonstationary transport equations in the diffusive limit, SIAM Journal on Numerical Analysis, 35 (1998), 1073-1094.  doi: 10.1137/S0036142996305558.

[31]

M. Lemou and L. Mieussens, A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit, SIAM Journal on Scientific Computing, 31 (2008), 334-368.  doi: 10.1137/07069479X.

[32]

Q. Li and L. Wang, Uniform regularity for linear kinetic equations with random input based on hypocoercivity, SIAM/ASA Journal on Uncertainty Quantification, 5 (2017), 1-20.  doi: 10.1137/16M1106675.

[33]

P. L. Lions and G. Toscani, Diffusive limit for finite velocity Boltzmann kinetic models, Revista Matematica Iberoamericana, 13 (1997), 473-513.  doi: 10.4171/RMI/228.

[34]

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 Modeling & Simulation, 16 (2018), 1085-1114.  doi: 10.1137/17M1151730.

[35]

T. Platkowski and R. Illner, Discrete velocity models of the Boltzmann equation: A survey on the mathematical aspects of the theory, SIAM Review, 30 (1988), 213-255.  doi: 10.1137/1030045.

[36]

A. Pulvirenti and G. Toscani, Fast diffusion as a limit of a two-velocity kinetic model, Circ. Mat. Palermo Suppl, 45 (1996), 521-528. 

[37]

F. Salvarani and G. Toscani, The diffusive limit of Carleman-type models in the range of very fast diffusion equations, Journal of Evolution Equations, 9 (2009), 67-80.  doi: 10.1007/s00028-009-0005-y.

[38]

F. Salvarani and J. J. L. Vázquez, The diffusive limit for Carleman-type kinetic models, Nonlinearity, 18 (2005), 1223-1248.  doi: 10.1088/0951-7715/18/3/015.

[39]

Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Mathematical Journal, 14 (1985), 249-275.  doi: 10.14492/hokmj/1381757663.

[40]

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: Mathematical Modelling and Numerical Analysis, 52 (2018), 1651-1678.  doi: 10.1051/m2an/2018024.

[41]

R. C. Smith, Uncertainty Quantification: Theory, Implementation, and Applications, vol. 12, Computational Science & Engineering, 12. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2014.

[42]

T. UmedaS. Kawashima and Y. Shizuta, On the decay of solutions to the linearized equations of electro-magneto-fluid dynamics, Japan Journal of Applied Mathematics, 1 (1984), 435-457.  doi: 10.1007/BF03167068.

[43] D. Xiu, Numerical Methods for Stochastic Computations: A Spectral Method Approach, Princeton University Press, 2010. 
[44]

D. Xiu and G. E. Karniadakis, The Wiener–Askey polynomial chaos for stochastic differential equations, SIAM journal on Scientific Computing, 24 (2002), 619-644.  doi: 10.1137/S1064827501387826.

show all references

References:
[1]

A. BarthC. Schwab and N. Zollinger, Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients, Numerische Mathematik, 119 (2011), 123-161.  doi: 10.1007/s00211-011-0377-0.

[2]

A. Bellouquid, A diffusive limit for nonlinear discrete velocity models, Mathematical Models and Methods in Applied Sciences, 13 (2003), 35-58.  doi: 10.1142/S0218202503002374.

[3]

J. E. Broadwell, Shock structure in a simple discrete velocity gas, The Physics of Fluids, 7 (1964), 1243-1247.  doi: 10.1063/1.1711368.

[4]

T. Carleman, Problemes Mathématiques dans la Théorie Cinétique de Gaz, vol. 2, Almqvist & Wiksell, 1957.

[5]

J. CharrierR. Scheichl and A. L. Teckentrup, Finite element error analysis of elliptic PDEs with random coefficients and its application to multilevel Monte Carlo methods, SIAM Journal on Numerical Analysis, 51 (2013), 322-352.  doi: 10.1137/110853054.

[6]

S. Chen and G. D. Doolen, Lattice boltzmann method for fluid flows, Annual Review of Fluid Mechanics, 30 (1998), 329-364.  doi: 10.1146/annurev.fluid.30.1.329.

[7]

C. Villani, Hypocoercivity, American Mathematical Soc., 2009. doi: 10.1090/S0065-9266-09-00567-5.

[8]

F. Golse, S. Jin and C. D. Levermore, The convergence of numerical transfer schemes in diffusive regimes Ⅰ: Discrete-ordinate method, SIAM Journal on Numerical Analysis, 36 (1999), 1333–1369, http://epubs.siam.org/doi/abs/10.1137/S0036142997315986. doi: 10.1137/S0036142997315986.

[9]

S.-Y. Ha and A. E. Tzavaras, Lyapunov functionals and l1-stability for discrete velocity boltzmann equations, Communications in Mathematical Physics, 239 (2003), 65-92.  doi: 10.1007/s00220-003-0866-9.

[10]

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.

[11]

J. Hu and S. Jin, A Stochastic Galerkin method for the Boltzmann equation with uncertainty, Journal of Computational Physics, 315 (2016), 150–168, http://dx.doi.org/10.1016/j.jcp.2016.03.047. doi: 10.1016/j.jcp.2016.03.047.

[12]

J. Hu and S. Jin, Uncertainty quantification for kinetic equations, Uncertainty Quantification for Hyperbolic and Kinetic Equations, 14 (2018), 193-229.  doi: 10.1007/978-3-319-67110-9_6.

[13]

R. Illner and M. C. Reed, The decay of solution of the Carleman model, Mathematical Method in the Applied Sciences, 3 (1981), 121-127.  doi: 10.1002/mma.1670030110.

[14]

R. Illner and M. C. Reed, Decay to equilibrium for the Carleman model in a box, SIAM Journal on Applied Mathematics, 44 (1984), 1067-1075.  doi: 10.1137/0144076.

[15]

K. Inoue and T. Nishida, On the Broadwell model of the Boltzmann equation for a simple discrete velocity gas, Applied Mathematics and Optimization, 3 (1976), 27-49.  doi: 10.1007/BF02106189.

[16]

S. Jin, Efficient asymptotic-preserving (ap) schemes for some multiscale kinetic equations, SIAM Journal on Scientific Computing, 21 (1999), 441-454.  doi: 10.1137/S1064827598334599.

[17]

S. Jin, Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review, Lecture Notes for Summer School on Methods and Models of Kinetic Theory (M & MKT), Porto Ercole (Grosseto, Italy), 3 (2012), 177–216.

[18]

S. Jin, J.-G. J. 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 acro decomposition-based asymptotic-preserving method, Res Math Sci, 4 (2017), Paper No. 15, 25 pp. doi: 10.1186/s40687-017-0105-1.

[19]

S. Jin and L. Liu, An asymptotic-preserving stochastic galerkin method for the semiconductor boltzmann equation with random inputs and diffusive scalings, Siam Journal on Multiscale Model & Simulation, 15 (2017), 157-183.  doi: 10.1137/15M1053463.

[20]

S. JinL. Pareschi and G. Toscani, Diffusive relaxation schemes for multiscale discrete-velocity kinetic equations, SIAM Journal on Numerical Analysis, 35 (1998), 2405-2439.  doi: 10.1137/S0036142997315962.

[21]

S. JinD. Xiu and X. Zhu, Asymptotic-preserving methods for hyperbolic and transport equations with random inputs and diffusive scalings, Journal of Computational Physics, 289 (2015), 35-52.  doi: 10.1016/j.jcp.2015.02.023.

[22]

S. Jin and Y. Zhu, Hypocoercivity and uniform regularity for the vlasov–poisson–fokker–planck system with uncertainty and multiple scales, SIAM Journal on Mathematical Analysis, 50 (2018), 1790-1816.  doi: 10.1137/17M1123845.

[23]

S. Kawashima, Global existence and stability of solutions for discrete velocity models of the Boltzmann equation, North-Holland Mathematics Studies, 98 (1984), 59–85, https://www.sciencedirect.com/science/article/pii/S0304020808714920. doi: 10.1016/S0304-0208(08)71492-0.

[24]

S. Kawashima, Smooth global solutions for two-dimensional equations of electro-magneto-fluid dynamics, Japan Journal of Applied Mathematics, 1 (1984), 207-222.  doi: 10.1007/BF03167869.

[25]

S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, PhD thesis, Kyoto University, 1984.

[26]

S. Kawashima, Large-time Behavior of Solutions of the Discrete Boltzmann Equation, Physics, 589 (1987), 563-589.  doi: 10.1007/BF01208958.

[27]

S. Kawashima, The boltzmann equation and thirteen moments, Japan Journal of Applied Mathematics, 7 (1990), 301-320.  doi: 10.1007/BF03167846.

[28]

S. KawashimaA. Matsumura and T. Nishida, On the fluid-dynamical approximation to the Boltzmann equation at the level of the Navier-Stokes equation, Communications in Mathematical Physics, 70 (1979), 97-124.  doi: 10.1007/BF01982349.

[29]

S. KawashimaM. Okada and Ot hers, Smooth global solutions for the one-dimensional equations in magnetohydrodynamics, Proceedings of the Japan Academy, Series A, Mathematical Sciences, 58 (1982), 384-387.  doi: 10.3792/pjaa.58.384.

[30]

A. Klar, An asymptotic-induced scheme for nonstationary transport equations in the diffusive limit, SIAM Journal on Numerical Analysis, 35 (1998), 1073-1094.  doi: 10.1137/S0036142996305558.

[31]

M. Lemou and L. Mieussens, A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit, SIAM Journal on Scientific Computing, 31 (2008), 334-368.  doi: 10.1137/07069479X.

[32]

Q. Li and L. Wang, Uniform regularity for linear kinetic equations with random input based on hypocoercivity, SIAM/ASA Journal on Uncertainty Quantification, 5 (2017), 1-20.  doi: 10.1137/16M1106675.

[33]

P. L. Lions and G. Toscani, Diffusive limit for finite velocity Boltzmann kinetic models, Revista Matematica Iberoamericana, 13 (1997), 473-513.  doi: 10.4171/RMI/228.

[34]

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 Modeling & Simulation, 16 (2018), 1085-1114.  doi: 10.1137/17M1151730.

[35]

T. Platkowski and R. Illner, Discrete velocity models of the Boltzmann equation: A survey on the mathematical aspects of the theory, SIAM Review, 30 (1988), 213-255.  doi: 10.1137/1030045.

[36]

A. Pulvirenti and G. Toscani, Fast diffusion as a limit of a two-velocity kinetic model, Circ. Mat. Palermo Suppl, 45 (1996), 521-528. 

[37]

F. Salvarani and G. Toscani, The diffusive limit of Carleman-type models in the range of very fast diffusion equations, Journal of Evolution Equations, 9 (2009), 67-80.  doi: 10.1007/s00028-009-0005-y.

[38]

F. Salvarani and J. J. L. Vázquez, The diffusive limit for Carleman-type kinetic models, Nonlinearity, 18 (2005), 1223-1248.  doi: 10.1088/0951-7715/18/3/015.

[39]

Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Mathematical Journal, 14 (1985), 249-275.  doi: 10.14492/hokmj/1381757663.

[40]

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: Mathematical Modelling and Numerical Analysis, 52 (2018), 1651-1678.  doi: 10.1051/m2an/2018024.

[41]

R. C. Smith, Uncertainty Quantification: Theory, Implementation, and Applications, vol. 12, Computational Science & Engineering, 12. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2014.

[42]

T. UmedaS. Kawashima and Y. Shizuta, On the decay of solutions to the linearized equations of electro-magneto-fluid dynamics, Japan Journal of Applied Mathematics, 1 (1984), 435-457.  doi: 10.1007/BF03167068.

[43] D. Xiu, Numerical Methods for Stochastic Computations: A Spectral Method Approach, Princeton University Press, 2010. 
[44]

D. Xiu and G. E. Karniadakis, The Wiener–Askey polynomial chaos for stochastic differential equations, SIAM journal on Scientific Computing, 24 (2002), 619-644.  doi: 10.1137/S1064827501387826.

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