Skip to main content

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

  • Ruiwen Shu (a1), Jingwei Hu (a2) and Shi Jin (a1) (a3)

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.

Corresponding author
*Corresponding author. Email addresses: (R. Shu), (J. Hu), (S. Jin)
Hide All
[1] Alpert, B., A class of bases in L2 for the sparse representation of integral operators, SIAM J. Math. Anal., 24 (1993), pp. 246262.
[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. 10051034.
[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. 800825.
[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.
[5] Bird, G. A., Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Clarendon Press, Oxford, 1994.
[6] Bobylev, A. V., One class of invariant solutions of the Boltzmann equation, Akademiia Nauk SSSR, Doklady, 231 (1976), pp. 571574.
[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. 4761.
[8] Bungartz, H. J. and Griebel, M., Sparse grids, Acta Numerica, 13 (2004), pp. 147269.
[9] Cercignani, C., The Boltzmann Equation and Its Applications, Springer-Verlag, New York, 1988.
[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. 76257648.
[11] Garcke, J. and Griebel, M., Sparse Grids and Applications, Springer, 2013.
[12] Ghanem, R. G. and Spanos, P. D., Stochastic Finite Elements: A Spectral Approach, Springer-Verlag, New York, 1991.
[13] Griebel, M., Adaptive sparse grid multilevel methods for elliptic PDEs based on finite differences, Computing, 61 (1998), pp. 151179.
[14] Griebel, M. and Zumbusch, G., Adaptive sparse grids for hyperbolic conservation laws, in Hyperbolic Problems: Theory, Numerics, Applications, Springer, 1999, pp. 411422.
[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.
[16] Hu, J. and Jin, S., A stochastic Galerkin method for the Boltzmann equation with uncertainty, J. Comput. Phys., 315 (2016), pp. 150168.
[17] Krook, M. and Wu, T. T., Formation of Maxwellian tails, Phys. Fluids, 20 (1977), pp. 15891595.
[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. 391427, 429–461.
[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.
[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. 502531.
[21] Mouhot, C. and Pareschi, L., Fast algorithms for computing the Boltzmann collision operator, Math. Comput., 75 (2006), pp. 18331852.
[22] Narayan, A. and Zhou, T., Stochastic collocation on unstructured multivariate meshes, Commun. Comput. Phys., 18 (2015), pp. 136.
[23] Niederreiter, H., Hellekalek, P., Larcher, G. and Zinterhof, P., Monte Carlo and Quasi-Monte Carlo Methods 1996, Springer-Verlag, 1998.
[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. 23092345.
[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.
[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. 777819.
[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. 32283250.
[28] Smolyak, S., Quadrature and interpolation formulas for tensor products of certain classes of functions, Doklady Akademii Nauk SSSR, 4 (1963), pp. 240243.
[29] Wang, Z., Tang, Q., Guo, W. and Cheng, Y., Sparse grid discontinuous Galerkin methods for high-dimensional elliptic equations, J. Comput. Phys., accepted.
[30] Xiu, D., Fast numerical methods for stochastic computations: a review, Commun. Comput. Phys., 5 (2009), pp. 242272.
[31] Xiu, Dongbin, Numerical Methods for Stochastic Computation, Princeton University Press, Princeton, New Jersey, 2010.
[32] Xiu, D. and Hesthaven, J., High-order collocation methods for differential equations with random inputs, SIAM J. Sci. Comput., 27 (2005), pp. 11181139.
[33] Zenger, C., Sparse grids, in Parallel Algorithms for Partial Differential Equations, Proceedings of the Sixth GAMM-Seminar, vol. 31, 1990.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Numerical Mathematics: Theory, Methods and Applications
  • ISSN: 1004-8979
  • EISSN: 2079-7338
  • URL: /core/journals/numerical-mathematics-theory-methods-and-applications
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed