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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. Bardos, R. 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. [2] C. Cercignani, The Boltzmann Equation and Its Applications, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9. [3] Z. Chen, L. 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. [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. [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. [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. [7] F. Golse, S. 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. [8] D. Gottlieb and D. Xiu, Galerkin method for wave equations with uncertain coefficients, Comm Comput. Phys, 3 (2008), 505-518. [9] J.O. Hirschfelder, R.B. Bird and E.L. Spotz, The transport properties for non-polar gases, J. Chem. Phys., 16 (1948), 968-981. [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. [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. [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. [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. [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. [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. [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. [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. [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. [19] S. Jin, L. Pareschi and T. Giuseppe, Uniformly accurate diffusive relaxation schemes for multiscale transport equations, SIAM J. Num., 38 (2000), 913-936. doi: 10.1137/S0036142998347978. [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. [21] S. Jin, D. 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. [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. [23] A. Jüngel, Transport Equations for Semiconductors, Springer, 2009. doi: 10.1007/978-3-540-89526-8. [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. [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. [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. [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. [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, 201). [29] P. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2. [30] E. Poupaud, Diffusion approximation of the linear semiconductor boltzmann equation: Analysis of boundary layers, Asymptot. Anal, 4 (1991), 293-317. [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. [32] C. Ringhofer, C. 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. [33] R. Shu, J. 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. [34] D. Xiu, Numerical Methods for Stochastic Computations, Princeton University Press, Princeton, New Jersey, 2010. [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. [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.

show all references

##### References:
 [1] C. Bardos, R. 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. [2] C. Cercignani, The Boltzmann Equation and Its Applications, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9. [3] Z. Chen, L. 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. [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. [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. [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. [7] F. Golse, S. 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. [8] D. Gottlieb and D. Xiu, Galerkin method for wave equations with uncertain coefficients, Comm Comput. Phys, 3 (2008), 505-518. [9] J.O. Hirschfelder, R.B. Bird and E.L. Spotz, The transport properties for non-polar gases, J. Chem. Phys., 16 (1948), 968-981. [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. [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. [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. [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. [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. [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. [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. [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. [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. [19] S. Jin, L. Pareschi and T. Giuseppe, Uniformly accurate diffusive relaxation schemes for multiscale transport equations, SIAM J. Num., 38 (2000), 913-936. doi: 10.1137/S0036142998347978. [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. [21] S. Jin, D. 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. [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. [23] A. Jüngel, Transport Equations for Semiconductors, Springer, 2009. doi: 10.1007/978-3-540-89526-8. [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. [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. [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. [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. [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, 201). [29] P. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2. [30] E. Poupaud, Diffusion approximation of the linear semiconductor boltzmann equation: Analysis of boundary layers, Asymptot. Anal, 4 (1991), 293-317. [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. [32] C. Ringhofer, C. 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. [33] R. Shu, J. 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. [34] D. Xiu, Numerical Methods for Stochastic Computations, Princeton University Press, Princeton, New Jersey, 2010. [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. [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.
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