Hostname: page-component-89b8bd64d-r6c6k Total loading time: 0 Render date: 2026-05-07T23:19:39.907Z Has data issue: false hasContentIssue false
  • English
  • Français

A Stochastic Galerkin Method for the Boltzmann Equation with Multi-Dimensional Random Inputs Using Sparse Wavelet Bases

Part of: Numerical analysis: Ordinary differential equations Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  09 May 2017

Ruiwen Shu ,
Jingwei Hu  and
Shi Jin
Ruiwen Shu*
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, USA
Jingwei Hu*
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
Shi Jin*
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, USA Institute of Natural Sciences, School of Mathematical Science, MOE-LSEC and SHL-MAC, Shanghai Jiao Tong University, Shanghai 200240, China
*
*Corresponding author. Email addresses:rshu2@math.wisc.edu (R. Shu), jingweihu@purdue.edu (J. Hu), sjin@wisc.edu (S. Jin)
*Corresponding author. Email addresses:rshu2@math.wisc.edu (R. Shu), jingweihu@purdue.edu (J. Hu), sjin@wisc.edu (S. Jin)
*Corresponding author. Email addresses:rshu2@math.wisc.edu (R. Shu), jingweihu@purdue.edu (J. Hu), sjin@wisc.edu (S. Jin)
Get access

Abstract

We propose a stochastic Galerkin method using sparse wavelet bases for the Boltzmann equation with multi-dimensional random inputs. Themethod uses locally supported piecewise polynomials as an orthonormal basis of the random space. By a sparse approach, only a moderate number of basis functions is required to achieve good accuracy in multi-dimensional random spaces. We discover a sparse structure of a set of basis-related coefficients, which allows us to accelerate the computation of the collision operator. Regularity of the solution of the Boltzmann equation in the random space and an accuracy result of the stochastic Galerkin method are proved in multi-dimensional cases. The efficiency of the method is illustrated by numerical examples with uncertainties from the initial data, boundary data and collision kernel.

Keywords

Uncertainty quantificationBoltzmann equationstochastic Galerkin methodssparse wavelets

MSC classification

Secondary: 35Q20: Boltzmann equations 65L60: Finite elements, Rayleigh-Ritz, Galerkin and collocation methods

Information

Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 2 , May 2017 , pp. 465 - 488
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1] Alpert, B., A class of bases in L2 for the sparse representation of integral operators, SIAM J. Math. Anal., 24 (1993), pp. 246–262.Google Scholar
[2] Babuska, I., Nobile, F. and Tempone, R., A stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Numer. Anal., 45 (2007), pp. 1005–1034.CrossRefGoogle Scholar
[3] Babuska, I., Tempone, R. and Zouraris, G. E., Galerkin finite element approximations of stochastic elliptic partial differential equations, SIAM J. Numer. Anal., 42 (2004), pp. 800–825.Google Scholar
[4] Back, J., Nobile, F., Tamellini, L. and Tempone, R., Stochastic spectral Galerkin and collocation methods for PDEs with random coefficients: a numerical comparison, in Spectral and High Order Methods for Partial Differential Equations, Hesthaven, E. M. R. J. S., ed., Springer-Verlag Berlin Heidelberg, 2011.Google Scholar
[5] Bird, G. A., Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Clarendon Press, Oxford, 1994.Google Scholar
[6] Bobylev, A. V., One class of invariant solutions of the Boltzmann equation, Akademiia Nauk SSSR, Doklady, 231 (1976), pp. 571–574.Google Scholar
[7] Bouchut, F. and Desvillettes, L., A proof of the smoothing properties of the positive part of Boltzmann's kernel, Revista Matemática Iberoamericana, 14 (1998), pp. 47–61.CrossRefGoogle Scholar
[8] Bungartz, H. J. and Griebel, M., Sparse grids, Acta Numerica, 13 (2004), pp. 147–269.CrossRefGoogle Scholar
[9] Cercignani, C., The Boltzmann Equation and Its Applications, Springer-Verlag, New York, 1988.Google Scholar
[10] Filbet, F. and Jin, S., A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources, J. Comput. Phys., 229 (2010), pp. 7625–7648.Google Scholar
[11] Garcke, J. and Griebel, M., Sparse Grids and Applications, Springer, 2013.Google Scholar
[12] Ghanem, R. G. and Spanos, P. D., Stochastic Finite Elements: A Spectral Approach, Springer-Verlag, New York, 1991.Google Scholar
[13] Griebel, M., Adaptive sparse grid multilevel methods for elliptic PDEs based on finite differences, Computing, 61 (1998), pp. 151–179.Google Scholar
[14] Griebel, M. and Zumbusch, G., Adaptive sparse grids for hyperbolic conservation laws, in Hyperbolic Problems: Theory, Numerics, Applications, Springer, 1999, pp. 411–422.Google Scholar
[15] Guo, W. and Cheng, Y., A sparse grid discontinuous Galerkin method for high-dimensional transport equations and its application to kinetic simulations, SIAM J. Sci. Comput., accepted.Google Scholar
[16] Hu, J. and Jin, S., A stochastic Galerkin method for the Boltzmann equation with uncertainty, J. Comput. Phys., 315 (2016), pp. 150–168.CrossRefGoogle Scholar
[17] Krook, M. and Wu, T. T., Formation of Maxwellian tails, Phys. Fluids, 20 (1977), pp. 1589–1595.CrossRefGoogle Scholar
[18] Lions, P. L., Compactness in Boltzmann's equation via Fourier integral operators and applications. I, II, Journal of Mathematics of Kyoto University, 34 (1994), pp. 391–427, 429–461.Google Scholar
[19] Maître, O. P. L. and Knio, O. M., Spectral Methods for Uncertainty Quantification, Scientific Computation, with Applications to Computational Fluid Dynamics, Springer, New York, 2010.Google Scholar
[20] Maître, O. P. L., Najm, H. N., Ghanem, R. G. and Knio, O. M., Multi-resolution analysis of Wiener-type uncertainty propagation schemes, J. Comput. Phys., 197 (2004), pp. 502–531.Google Scholar
[21] Mouhot, C. and Pareschi, L., Fast algorithms for computing the Boltzmann collision operator, Math. Comput., 75 (2006), pp. 1833–1852.Google Scholar
[22] Narayan, A. and Zhou, T., Stochastic collocation on unstructured multivariate meshes, Commun. Comput. Phys., 18 (2015), pp. 1–36.Google Scholar
[23] Niederreiter, H., Hellekalek, P., Larcher, G. and Zinterhof, P., Monte Carlo and Quasi-Monte Carlo Methods 1996, Springer-Verlag, 1998.Google Scholar
[24] Nobile, F., Tempone, R. and Webster, C., A sparse grid stochastic collocation method for partial differential equations with random input data, SIAM J. Numer. Anal. 46 (2008), pp. 2309–2345.CrossRefGoogle Scholar
[25] Schiavazzi, D., Doostan, A. and Iaccarino, G., Sparse multiresolution stochastic approximation for uncertainty quantification, Recent Advances in Scientific Computing and Applications, 586 (2013), pp. 295.CrossRefGoogle Scholar
[26] Schwab, C., Süli, E. and Todor, R. A., Sparse finite element approximation of high-dimensional transport-dominated diffusion problems, ESAIM: Mathematical Modelling and Numerical Analysis, 42 (2008), pp. 777–819.CrossRefGoogle Scholar
[27] Shen, J. and Yu, H., Efficient spectral sparse grid methods and applications to high-dimensional elliptic problems, SIAM J. Sci. Comput., 32 (2010), pp. 3228–3250.CrossRefGoogle Scholar
[28] Smolyak, S., Quadrature and interpolation formulas for tensor products of certain classes of functions, Doklady Akademii Nauk SSSR, 4 (1963), pp. 240–243.Google Scholar
[29] Wang, Z., Tang, Q., Guo, W. and Cheng, Y., Sparse grid discontinuous Galerkin methods for high-dimensional elliptic equations, J. Comput. Phys., accepted.Google Scholar
[30] Xiu, D., Fast numerical methods for stochastic computations: a review, Commun. Comput. Phys., 5 (2009), pp. 242–272.Google Scholar
[31] Xiu, Dongbin, Numerical Methods for Stochastic Computation, Princeton University Press, Princeton, New Jersey, 2010.Google Scholar
[32] Xiu, D. and Hesthaven, J., High-order collocation methods for differential equations with random inputs, SIAM J. Sci. Comput., 27 (2005), pp. 1118–1139.CrossRefGoogle Scholar
[33] Zenger, C., Sparse grids, in Parallel Algorithms for Partial Differential Equations, Proceedings of the Sixth GAMM-Seminar, vol. 31, 1990.Google Scholar