High Order IMEX Stochastic Galerkin Schemes for Linear Transport Equation with Random Inputs and Diffusive Scalings

doi:10.1007/s42967-023-00249-x

Abstract

Abstract: In this paper, we consider the high order method for solving the linear transport equations under diffusive scaling and with random inputs. To tackle the randomness in the problem, the stochastic Galerkin method of the generalized polynomial chaos approach has been employed. Besides, the high order implicit-explicit scheme under the micro-macro decomposition framework and the discontinuous Galerkin method have been employed. We provide several numerical experiments to validate the accuracy and the stochastic asymptotic-preserving property.

Key words: Stochastic Galerkin scheme, linear transport equations, generalized polynomial approach, stochastic asymptotic-preserving property

Zheng Chen, Lin Mu. High Order IMEX Stochastic Galerkin Schemes for Linear Transport Equation with Random Inputs and Diffusive Scalings[J]. Communications on Applied Mathematics and Computation, 2024, 6(1): 325-339.

TrendMD

References

[1] Bennoune, M., Lemou, M., Mieussens, L.:Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics. J. Comput. Phys. 227(8), 3781-3803 (2008)
[2] Carrillo, J.A., Goudon, T., Lafitte, P., Vecil, F.:Numerical schemes of diffusion asymptotics and moment closures for kinetic equations. J. Sci. Comput. 36, 113-149 (2008)
[3] Chen, Y., Chen, Z., Cheng, Y., Gillman, A., Li, F.:Study of discrete scattering operators for some linear kinetic models. In:Numerical Partial Differential Equations and Scientific Computing. pp. 99-136. Springer, New York (2016)
[4] Chen, Z., Hauck, C.D.:Multiscale convergence properties for spectral approximations of a model kinetic equation. Math. Comput. 88, 2257-2293 (2019)
[5] Chen, Z., Liu, L., Mu, L.:DG-IMEX stochastic Galerkin schemes for linear transport equation with random inputs and diffusive scalings. J. Sci. Comput. 73, 566-592 (2017)
[6] Chen, Z., Mu, L.:Solving the linear transport equation by a deep neural network approach. Discrete Contin. Dyn. Syst.-S 15(4), 669-686 (2022)
[7] Daus, E., Jin, S., Liu, L.:On the multi-species Boltzmann equation with uncertainty and its stochastic Galerkin approximation. ESAIM 55, 1323-1345 (2021)
[8] Escalante, J., Heitzinger, C.:Stochastic Galerkin methods for the Boltzmann-Poisson system. J. Comput. Phys. 466, 111400 (2022)
[9] Hu, J.:Jin, S.:Uncertainty quantification for kinetic equations. Uncertainty Quantification for Hyperbolic and Kinetic Equations, pp. 193-229. Springer, Cham. (2017)
[10] Jang, J., Li, F., Qiu, J., Xiong, T.:Analysis of asymptotic preserving DG-IMEX schemes for linear kinetic transport equations in a diffusive scaling. SIAM J. Numer. Anal. 52, 2048-2072 (2014)
[11] Jang, J., Li, F., Qiu, J., Xiong, T.:High order asymptotic preserving DG-IMEX schemes for discrete-velocity kinetic equations in a diffusive scaling. J. Comput. Phys. 281, 199-224 (2015)
[12] Jin, S.:Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations:a review. Riv. Math. Univ. Parma (N.S.) 3(2), 177-216 (2012)
[13] Jin, S.:Mathematical analysis and numerical methods for multiscale kinetic equations with uncertainties. Proc. Int. Cong. Math. 2018, 3611-3639 (2018)
[14] Jin, S.:Asymptotic-preserving schemes for multiscale physical problems. Acta Numerica 31, 415-489 (2022)
[15] Jin, S., Liu, J., Ma, Z.: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, 15 (2017)
[16] Jin, S., Lu, H., Pareschi, L.:Efficient stochastic asymptotic-preserving implicit-explicit methods for transport equations with diffusive scalings and random inputs. SIAM J. Sci. Comput. 40, A671-A696 (2018)
[17] Jin, S., Xiu, D., Zhu, X.:Asymptotic-preserving methods for hyperbolic and transport equations with random inputs and diffusive scalings. J. Comput. Phys. 289, 35-52 (2015)
[18] Klar, A.:An asymptotic-induced scheme for nonstationary transport equations in the diffusive limit. SIAM J. Numer. Anal. 35, 1073-1094 (1998)
[19] Lafitte, P., Samaey, G.:Asymptotic-preserving projective integration schemes for kinetic equations in the diffusion limit. SIAM J. Sci. Comput. 34, 579-602 (2012)
[20] Laiu, M., Chen, Z., Hauck, C.D.:A fast implicit solver for semiconductor models in one space dimension. J. Comput. Phys. 417, 109567 (2020)
[21] Lemou, M., Mieussens, L.:A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit. SIAM J. Sci. Comput. 31, 334-368 (2010)
[22] Liu, L.:Uniform spectral convergence of the stochastic Galerkin method for the linear semiconductor Boltzmann equation with random inputs and diffusive scalings. Kinet. Relat. Models 11(5), 1139-1156 (2018)
[23] Liu, L.:A stochastic asymptotic-preserving scheme for the bipolar semiconductor Boltzmann-Poisson system with random inputs and diffusive scalings. J. Comput. Phys. 376, 634-659 (2019)
[24] Liu, L.:A bi-fidelity DG-IMEX method for the linear transport equation with random parameters. In:ECCOMAS Congress 2020 (2021)
[25] Liu, L., Jin, S.: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, 1085-1114 (2018)
[26] Lowrie, R.B., Morel, J.E.:Methods for hyperbolic systems with stiff relaxation. Int. J. Numer. Methods Fluids 40, 413-23 (2002)
[27] Pareschi, L., Russo, G.:Efficient asymptotic preserving deterministic methods for the Boltzmann equation. Lecture Series held at the von Karman Institute, Rhode St. Gense, Belgium, AVT-194 RTO AVT/VKI, (2011)
[28] Peng, Z., Cheng, Y., Qiu, J.-M., Li, F.:Stability-enhanced AP IMEX-LDG schemes for linear kinetic transport equations under a diffusive scaling. J. Comput. Phys. 415, 109485 (2018)
[29] Reed, W.H., Hill, T.R.:Triangular Mesh Methods for the Neutron Transport Equation. Los Alamos Scientific Lab, New Mexico (USA) (1973)
[30] Shu, C.-W.:Discontinuous Galerkin methods:general approach and stability. In:Numerical Solutions of Partial Differential Equations. Advanced Mathematics Training Course, CRM Barcelona, pp. 149-201. Birkhäuser, Basel (2009)
[31] Shu, C.-W.:Discontinuous Galerkin method for time-dependent problems:survey and recent developments. In:Feng, X., Karakashian, O., Xing, Y. (eds.) Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations. The IMA Volumes in Mathematics and Its Applications, vol. 157. Springer, Cham. (2014)
[32] Xiong, T., Jang, J., Li, F., Qiu, J.:High order asymptotic preserving nodal discontinuous Galerkin IMEX schemes for the BGK equation. J. Comput. Phys. 284, 70-94 (2015)
[33] Xiu, D.:Numerical Methods for Stochastic Computations:a Spectral Method Approach. Princeton University Press, Princeton (2010)
[34] Xiu, D., Shen, J.:Efficient stochastic Galerkin methods for random diffusion equations. J. Comput. Phys. 228, 266-281 (2009)