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October  2018, 11(5): 1139-1156. doi: 10.3934/krm.2018044

Uniform spectral convergence of the stochastic Galerkin method for the linear semiconductor Boltzmann equation with random inputs and diffusive scaling

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

Received  June 2017 Revised  October 2017 Published  May 2018

Fund Project: The author was supported in part by Prof. Shi Jin’s NSF grants DMS-1522184 and DMS-1107291: RNMS KI-Net.

In this paper, we study the generalized polynomial chaos (gPC) based stochastic Galerkin method for the linear semiconductor Boltzmann equation under diffusive scaling and with random inputs from an anisotropic collision kernel and the random initial condition. While the numerical scheme and the proof of uniform-in-Knudsen-number regularity of the distribution function in the random space has been introduced in [15], the main goal of this paper is to first obtain a sharper estimate on the regularity of the solution-an exponential decay towards its local equilibrium, which then lead to the uniform spectral convergence of the stochastic Galerkin method for the problem under study.

Citation: Liu Liu. Uniform spectral convergence of the stochastic Galerkin method for the linear semiconductor Boltzmann equation with random inputs and diffusive scaling. Kinetic & Related Models, 2018, 11 (5) : 1139-1156. doi: 10.3934/krm.2018044
References:
[1]

C. BardosR. Santos and R. Sentis, Diffusion approximation and computation of the critical size, Trans. Amer. Math. Soc., 284 (1984), 617-649.  doi: 10.1090/S0002-9947-1984-0743736-0.  Google Scholar

[2]

C. Cercignani, The Boltzmann Equation and Its Applications, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9.  Google Scholar

[3]

Z. ChenL. Liu and L. Mu, DG-IMEX stochastic galerkin schemes for linear transport equation with random inputs and diffusive scalings, Journal of Scientific Computing, 73 (2017), 566-592.  doi: 10.1007/s10915-017-0439-2.  Google Scholar

[4]

N. Crouseilles, S. Jin, M. Lemou and L. Liu, Nonlinear geometric optics based multiscale stochastic galerkin methods for highly oscillatory transport equations with random inputs, preprint, 2017. Google Scholar

[5]

J. Deng, Implicit asymptotic preserving schemes for semiconductor boltzmann equation in the diffusive regime, International Journal of Numerical Analysis and Modeling, 11 (2014), 1-23.   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]

F. GolseS. Jin and C.D. Levermore, The convergence of numerical transfer schemes in diffusive regimes. Ⅰ. Discrete-ordinate method, SIAM J. Numer. Anal., 36 (1999), 1333-1369.  doi: 10.1137/S0036142997315986.  Google Scholar

[8]

D. Gottlieb and D. Xiu, Galerkin method for wave equations with uncertain coefficients, Comm Comput. Phys, 3 (2008), 505-518.   Google Scholar

[9]

J.O. HirschfelderR.B. Bird and E.L. Spotz, The transport properties for non-polar gases, J. Chem. Phys., 16 (1948), 968-981.   Google Scholar

[10]

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

[11]

_____, Uncertainty Quantification for Kinetic Equations, Uncertainty Quantification for Kinetic and Hyperbolic Equations, SEMA-SIMAI Springer Series, ed. S. Jin and L. Pareschi, to appear, 2017. Google Scholar

[12]

S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comp., 21 (1999), 441-454.  doi: 10.1137/S1064827598334599.  Google Scholar

[13]

_____, 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, 2010. Google Scholar

[14]

S. Jin, 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-macro decomposition based asymptotic preserving method, Research in Math. Sci., 4 (2017), Paper No. 15, 25 pp. doi: 10.1186/s40687-017-0105-1.  Google Scholar

[15]

S. Jin and L. Liu, An asymptotic-preserving stochastic Galerkin method for the semiconductor Boltzmann equation with random inputs and diffusive scalings, Multiscale Model. Simul., 15 (2017), 157-183.  doi: 10.1137/15M1053463.  Google Scholar

[16]

S. Jin and H. Lu, An asymptotic-preserving stochastic Galerkin method for the radiative heat transfer equations with random inputs and diffusive scalings, J. Comput. Phys., 334 (2017), 182-206.  doi: 10.1016/j.jcp.2016.12.033.  Google Scholar

[17]

S. Jin and Z. Ma, The discrete stochastic galerkin method for hyperbolic equations with non-smooth and random coefficeints, J. Sci. Comp, 74 (2018), 97-121.  doi: 10.1007/s10915-017-0426-7.  Google Scholar

[18]

S. Jin and L. Pareschi, Discretization of the multiscale semiconductor Boltzmann equation by diffusive relaxation schemes, J. Comput. Phys., 161 (2000), 312-330.  doi: 10.1006/jcph.2000.6506.  Google Scholar

[19]

S. JinL. Pareschi and T. Giuseppe, Uniformly accurate diffusive relaxation schemes for multiscale transport equations, SIAM J. Num., 38 (2000), 913-936.  doi: 10.1137/S0036142998347978.  Google Scholar

[20]

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: 10.1016/j.jcp.2017.01.059.  Google Scholar

[21]

S. JinD. Xiu and X. Zhu, Asymptotic-preserving methods for hyperbolic and transport equations with random inputs and diffusive scalings, J. Comp. Phys., 289 (2015), 25-52.  doi: 10.1016/j.jcp.2015.02.023.  Google Scholar

[22]

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: 10.1137/17M1123845.  Google Scholar

[23]

A. Jüngel, Transport Equations for Semiconductors, Springer, 2009. doi: 10.1007/978-3-540-89526-8.  Google Scholar

[24]

A. Klar, An asymptotic-induced scheme for nonstationary transport equations in the diffusive limit, SIAM J. Numer. Anal., 35 (1998), 1073-1094.  doi: 10.1137/S0036142996305558.  Google Scholar

[25]

O. P. Le Maître and O. M. Knio, Spectral Methods for Uncertainty Quantification, Springer, New York, 2010. doi: 10.1007/978-90-481-3520-2.  Google Scholar

[26]

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

[27]

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

[28]

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, preprint, (2017). Google Scholar

[29]

P. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2.  Google Scholar

[30]

E. Poupaud, Diffusion approximation of the linear semiconductor boltzmann equation: Analysis of boundary layers, Asymptot. Anal, 4 (1991), 293-317.   Google Scholar

[31]

A. K. Prinja, E. D. Fichtl and J. S. Warsa, Stochastic Methods for Uncertainty Quantification in Radiation Transport, In International Conference on Mathematics, Computational Methods & Reactor Physics, May 3-7, Saratoga Springs, New York, 2009. Google Scholar

[32]

C. RinghoferC. Schmeiser and A. Zwirchmayr, Moment methods for the semiconductor boltzmann equation on bounded position domains, SIAM J. Num. Anal., 39 (2001), 1078-1095.  doi: 10.1137/S0036142998335984.  Google Scholar

[33]

R. ShuJ. Hu and S. Jin, A stochastic Galerkin method for the Boltzmann equation with multi-dimensional random inputs using sparse wavelet bases, Numer. Math. Theory Methods Appl., 10 (2017), 465-488.  doi: 10.4208/nmtma.2017.s12.  Google Scholar

[34]

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

[35]

T. Zhou and T. Tang, Convergence analysis for spectral approximation to a scalar transport equation with a random wave speed, Journal of Computational Mathematics (ISSN: 0254-9409), 30 (2012), 643-656.  doi: 10.4208/jcm.1206-m4012.  Google Scholar

[36]

_____, Galerkin methods for stochastic hyperbolic problems using bi-orthogonal polynomials, J. Sci. Comput, 51 (2012), 274-292.  doi: 10.1007/s10915-011-9508-0.  Google Scholar

show all references

References:
[1]

C. BardosR. Santos and R. Sentis, Diffusion approximation and computation of the critical size, Trans. Amer. Math. Soc., 284 (1984), 617-649.  doi: 10.1090/S0002-9947-1984-0743736-0.  Google Scholar

[2]

C. Cercignani, The Boltzmann Equation and Its Applications, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9.  Google Scholar

[3]

Z. ChenL. Liu and L. Mu, DG-IMEX stochastic galerkin schemes for linear transport equation with random inputs and diffusive scalings, Journal of Scientific Computing, 73 (2017), 566-592.  doi: 10.1007/s10915-017-0439-2.  Google Scholar

[4]

N. Crouseilles, S. Jin, M. Lemou and L. Liu, Nonlinear geometric optics based multiscale stochastic galerkin methods for highly oscillatory transport equations with random inputs, preprint, 2017. Google Scholar

[5]

J. Deng, Implicit asymptotic preserving schemes for semiconductor boltzmann equation in the diffusive regime, International Journal of Numerical Analysis and Modeling, 11 (2014), 1-23.   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]

F. GolseS. Jin and C.D. Levermore, The convergence of numerical transfer schemes in diffusive regimes. Ⅰ. Discrete-ordinate method, SIAM J. Numer. Anal., 36 (1999), 1333-1369.  doi: 10.1137/S0036142997315986.  Google Scholar

[8]

D. Gottlieb and D. Xiu, Galerkin method for wave equations with uncertain coefficients, Comm Comput. Phys, 3 (2008), 505-518.   Google Scholar

[9]

J.O. HirschfelderR.B. Bird and E.L. Spotz, The transport properties for non-polar gases, J. Chem. Phys., 16 (1948), 968-981.   Google Scholar

[10]

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

[11]

_____, Uncertainty Quantification for Kinetic Equations, Uncertainty Quantification for Kinetic and Hyperbolic Equations, SEMA-SIMAI Springer Series, ed. S. Jin and L. Pareschi, to appear, 2017. Google Scholar

[12]

S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comp., 21 (1999), 441-454.  doi: 10.1137/S1064827598334599.  Google Scholar

[13]

_____, 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, 2010. Google Scholar

[14]

S. Jin, 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-macro decomposition based asymptotic preserving method, Research in Math. Sci., 4 (2017), Paper No. 15, 25 pp. doi: 10.1186/s40687-017-0105-1.  Google Scholar

[15]

S. Jin and L. Liu, An asymptotic-preserving stochastic Galerkin method for the semiconductor Boltzmann equation with random inputs and diffusive scalings, Multiscale Model. Simul., 15 (2017), 157-183.  doi: 10.1137/15M1053463.  Google Scholar

[16]

S. Jin and H. Lu, An asymptotic-preserving stochastic Galerkin method for the radiative heat transfer equations with random inputs and diffusive scalings, J. Comput. Phys., 334 (2017), 182-206.  doi: 10.1016/j.jcp.2016.12.033.  Google Scholar

[17]

S. Jin and Z. Ma, The discrete stochastic galerkin method for hyperbolic equations with non-smooth and random coefficeints, J. Sci. Comp, 74 (2018), 97-121.  doi: 10.1007/s10915-017-0426-7.  Google Scholar

[18]

S. Jin and L. Pareschi, Discretization of the multiscale semiconductor Boltzmann equation by diffusive relaxation schemes, J. Comput. Phys., 161 (2000), 312-330.  doi: 10.1006/jcph.2000.6506.  Google Scholar

[19]

S. JinL. Pareschi and T. Giuseppe, Uniformly accurate diffusive relaxation schemes for multiscale transport equations, SIAM J. Num., 38 (2000), 913-936.  doi: 10.1137/S0036142998347978.  Google Scholar

[20]

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: 10.1016/j.jcp.2017.01.059.  Google Scholar

[21]

S. JinD. Xiu and X. Zhu, Asymptotic-preserving methods for hyperbolic and transport equations with random inputs and diffusive scalings, J. Comp. Phys., 289 (2015), 25-52.  doi: 10.1016/j.jcp.2015.02.023.  Google Scholar

[22]

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: 10.1137/17M1123845.  Google Scholar

[23]

A. Jüngel, Transport Equations for Semiconductors, Springer, 2009. doi: 10.1007/978-3-540-89526-8.  Google Scholar

[24]

A. Klar, An asymptotic-induced scheme for nonstationary transport equations in the diffusive limit, SIAM J. Numer. Anal., 35 (1998), 1073-1094.  doi: 10.1137/S0036142996305558.  Google Scholar

[25]

O. P. Le Maître and O. M. Knio, Spectral Methods for Uncertainty Quantification, Springer, New York, 2010. doi: 10.1007/978-90-481-3520-2.  Google Scholar

[26]

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

[27]

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

[28]

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, preprint, (2017). Google Scholar

[29]

P. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2.  Google Scholar

[30]

E. Poupaud, Diffusion approximation of the linear semiconductor boltzmann equation: Analysis of boundary layers, Asymptot. Anal, 4 (1991), 293-317.   Google Scholar

[31]

A. K. Prinja, E. D. Fichtl and J. S. Warsa, Stochastic Methods for Uncertainty Quantification in Radiation Transport, In International Conference on Mathematics, Computational Methods & Reactor Physics, May 3-7, Saratoga Springs, New York, 2009. Google Scholar

[32]

C. RinghoferC. Schmeiser and A. Zwirchmayr, Moment methods for the semiconductor boltzmann equation on bounded position domains, SIAM J. Num. Anal., 39 (2001), 1078-1095.  doi: 10.1137/S0036142998335984.  Google Scholar

[33]

R. ShuJ. Hu and S. Jin, A stochastic Galerkin method for the Boltzmann equation with multi-dimensional random inputs using sparse wavelet bases, Numer. Math. Theory Methods Appl., 10 (2017), 465-488.  doi: 10.4208/nmtma.2017.s12.  Google Scholar

[34]

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

[35]

T. Zhou and T. Tang, Convergence analysis for spectral approximation to a scalar transport equation with a random wave speed, Journal of Computational Mathematics (ISSN: 0254-9409), 30 (2012), 643-656.  doi: 10.4208/jcm.1206-m4012.  Google Scholar

[36]

_____, Galerkin methods for stochastic hyperbolic problems using bi-orthogonal polynomials, J. Sci. Comput, 51 (2012), 274-292.  doi: 10.1007/s10915-011-9508-0.  Google Scholar

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