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 & 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

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

[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. Google Scholar

[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. Google Scholar

[7]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. 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 Journal on Mathematical Analysis, 50 (2018), 1790-1816. doi: 10.1137/17M1123845. Google Scholar

[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. Google Scholar

[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. Google Scholar

[25]

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

[26]

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

[27]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

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

[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. Google Scholar

[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. Google Scholar

[7]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. 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 Journal on Mathematical Analysis, 50 (2018), 1790-1816. doi: 10.1137/17M1123845. Google Scholar

[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. Google Scholar

[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. Google Scholar

[25]

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

[26]

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

[27]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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

[1]

Alexander Bobylev, Mirela Vinerean, Åsa Windfäll. Discrete velocity models of the Boltzmann equation and conservation laws. Kinetic & Related Models, 2010, 3 (1) : 35-58. doi: 10.3934/krm.2010.3.35

[2]

Niclas Bernhoff. Boundary layers for discrete kinetic models: Multicomponent mixtures, polyatomic molecules, bimolecular reactions, and quantum kinetic equations. Kinetic & Related Models, 2017, 10 (4) : 925-955. doi: 10.3934/krm.2017037

[3]

Davide Bellandi. On the initial value problem for a class of discrete velocity models. Mathematical Biosciences & Engineering, 2017, 14 (1) : 31-43. doi: 10.3934/mbe.2017003

[4]

Seung-Yeal Ha, Mitsuru Yamazaki. $L^p$-stability estimates for the spatially inhomogeneous discrete velocity Boltzmann model. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 353-364. doi: 10.3934/dcdsb.2009.11.353

[5]

Kazuo Aoki, Ansgar Jüngel, Peter A. Markowich. Small velocity and finite temperature variations in kinetic relaxation models. Kinetic & Related Models, 2010, 3 (1) : 1-15. doi: 10.3934/krm.2010.3.1

[6]

Seung-Yeal Ha, Shi Jin, Jinwook Jung. A local sensitivity analysis for the kinetic Kuramoto equation with random inputs. Networks & Heterogeneous Media, 2019, 14 (2) : 317-340. doi: 10.3934/nhm.2019013

[7]

Michael Herty, Lorenzo Pareschi, Mohammed Seaïd. Enskog-like discrete velocity models for vehicular traffic flow. Networks & Heterogeneous Media, 2007, 2 (3) : 481-496. doi: 10.3934/nhm.2007.2.481

[8]

Jean Dolbeault. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 361-380. doi: 10.3934/dcds.2002.8.361

[9]

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

[10]

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

[11]

Yanqing Wang. A semidiscrete Galerkin scheme for backward stochastic parabolic differential equations. Mathematical Control & Related Fields, 2016, 6 (3) : 489-515. doi: 10.3934/mcrf.2016013

[12]

N. Bellomo, A. Bellouquid. From a class of kinetic models to the macroscopic equations for multicellular systems in biology. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 59-80. doi: 10.3934/dcdsb.2004.4.59

[13]

Cliburn Chan, Andrew J.T. George, Jaroslav Stark. T cell sensitivity and specificity - kinetic proofreading revisited. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 343-360. doi: 10.3934/dcdsb.2003.3.343

[14]

Jianquan Li, Zhien Ma, Fred Brauer. Global analysis of discrete-time SI and SIS epidemic models. Mathematical Biosciences & Engineering, 2007, 4 (4) : 699-710. doi: 10.3934/mbe.2007.4.699

[15]

Alexander Alekseenko, Jeffrey Limbacher. Evaluating high order discontinuous Galerkin discretization of the Boltzmann collision integral in $ \mathcal{O}(N^2) $ operations using the discrete fourier transform. Kinetic & Related Models, 2019, 12 (4) : 703-726. doi: 10.3934/krm.2019027

[16]

Frederic Mazenc, Gonzalo Robledo, Michael Malisoff. Stability and robustness analysis for a multispecies chemostat model with delays in the growth rates and uncertainties. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1851-1872. doi: 10.3934/dcdsb.2018098

[17]

Martina Conte, Maria Groppi, Giampiero Spiga. Qualitative analysis of kinetic-based models for tumor-immune system interaction. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2393-2414. doi: 10.3934/dcdsb.2018060

[18]

Yanzhao Cao, Li Yin. Spectral Galerkin method for stochastic wave equations driven by space-time white noise. Communications on Pure & Applied Analysis, 2007, 6 (3) : 607-617. doi: 10.3934/cpaa.2007.6.607

[19]

Yayun Zheng, Xu Sun. Governing equations for Probability densities of stochastic differential equations with discrete time delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3615-3628. doi: 10.3934/dcdsb.2017182

[20]

Guy Métivier, K. Zumbrun. Existence and sharp localization in velocity of small-amplitude Boltzmann shocks. Kinetic & Related Models, 2009, 2 (4) : 667-705. doi: 10.3934/krm.2009.2.667

2018 Impact Factor: 1.38

Metrics

  • PDF downloads (24)
  • HTML views (89)
  • Cited by (0)

Other articles
by authors

[Back to Top]