Skip to main content
SpringerLink
Log in
Book cover

XVI International Conference on Hyperbolic Problems: Theory, Numerics, Applications

HYP 2016: Theory, Numerics and Applications of Hyperbolic Problems II pp 179–193Cite as

On Quantifying Uncertainties for the Linearized BGK Kinetic Equation

  • Conference paper
  • First Online:

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 237))

Abstract

We consider the linearized BGK equation and want to quantify uncertainties in the case of modeling errors. More specifically, we want to quantify the error produced if the predetermined equilibrium function is chosen inaccurately. In this paper, we consider perturbations in the velocity and in the temperature of the equilibrium function and consider how much the error is amplified in the solution.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   249.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   249.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    for practical purpose, the range of z is controlled by \(\varDelta t\) but we study the general case here

References

  1. G. Albi, L. Pareschi, M. Zanella, Uncertainty quantification in control problems for flocking models. Math. Probl. Eng. 850124, 14 (2015)

    Google Scholar 

  2. M. Branicki, A.J. Majda, Fundamental limitations of polynomial chaos for uncertainty quantification in systems with intermittent instabilities. Commun. Math. Sci. 11(1), 55–103 (2013)

    Article  MathSciNet  Google Scholar 

  3. I. Babuška, F. Nobile, R. Tempone, Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42(2), 800–825 (2004)

    Article  MathSciNet  Google Scholar 

  4. I. Babuška, F. Nobile, R. Tempone, A stochastic collocation method for elliptic partial differential equation with random input data. SIAM J. Numer. Anal. 45(3), 1005–1034 (2007)

    Article  MathSciNet  Google Scholar 

  5. I. Babuška, F. Nobile, R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45(3), 1005–1034 (2007)

    Article  MathSciNet  Google Scholar 

  6. A. Barth, C. Schwab, N. Zollinger, Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients. Numer. Math. 119(1), 123–161 (2011)

    Article  MathSciNet  Google Scholar 

  7. J. Charrier, R. Scheichl, A.L. Teckentrup, Finite element error analysis of elliptic PDEs with random coefficients and its application to multilevel Monte Carlo methods. SIAM J. Numer. Anal. 51(1), 322–352 (2013)

    Article  MathSciNet  Google Scholar 

  8. A. Chkifa, A. Cohen, C. Schwab, Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs. Journal de Mathématiques Pures et Appliquées (2014)

    Google Scholar 

  9. A. Cohen, R. DeVore, C. Schwab, Convergence rates of best N-term Galerkin approximations for a class of elliptic sPDEs. Found. Comput. Math. 10(6), 615–646 (2010)

    Article  MathSciNet  Google Scholar 

  10. A. Cohen, R. Devore, C. Schwab, Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDE’s. Anal. Appl. 9(01), 11–47 (2011)

    Article  MathSciNet  Google Scholar 

  11. B. Despres, B. Perthame, Uncertainty propagation; intrusive kinetic formulations of scalar conservation laws. SIAM/ASA J. Uncertain. Quantif. 4(1), 980–1013 (2016)

    Article  MathSciNet  Google Scholar 

  12. D. Xiu, G. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 (2002)

    Article  MathSciNet  Google Scholar 

  13. G. Fishman, Monte Carlo: Concepts, Algorithms, and Applications (Springer, New York, 2013)

    MATH  Google Scholar 

  14. M.B. Giles, Multilevel Monte Carlo path simulation. Op. Res. 56(3), 607–617 (2008)

    Article  MathSciNet  Google Scholar 

  15. R.G. Ghanem, A. Doostan, On the construction and analysis of stochastic models: characterization and propagation of the errors associated with limited data. J. Comput. Phys. 217(1), 63–81 (2006)

    Article  MathSciNet  Google Scholar 

  16. R.G. Ghanem, R.M. Kruger, Numerical solution of spectral stochastic finite element systems. Comput. Methods Appl. Mech. Eng. 129(3), 289–303 (1996)

    Article  Google Scholar 

  17. J. Hu, S. Jin, A stochastic Galerkin method for the Boltzmann equation with uncertainty. J. Comput. Phys. 315, 150–168 (2016)

    Article  MathSciNet  Google Scholar 

  18. Y.T. Hou, Q. Li, P. Zhang, Exploring the locally low dimensional structure in solving random elliptic PDEs. SIAM Multiscale Model. Simul. (2016)

    Google Scholar 

  19. Y.T. Hou, Q. Li, P. Zhang, A sparse decomposition of low rank symmetric positive semi-definite matrices. SIAM Multiscale Model. Simul. (2016)

    Google Scholar 

  20. S. Jin, L. Liu, An asymptotic-preserving stochastic Galerkin method for the semiconductor Boltzmann equation with random inputs and diffusive scalings. SIAM Multiscale Model. Simul. (2016)

    Google Scholar 

  21. S. Jin, H. Lu, An Asymptotic-Preserving stochastic Galerkin method for the radiative heat transfer equations with random inputs and diffusive scalings (2016)

    Google Scholar 

  22. S. Jin, Y. Zhu, The Vlasov-Poisson-Fokker-Planck system with uncertainty and a one-dimensional asymptotic-preserving method (2016)

    Google Scholar 

  23. S. Jin, D. Xiu, X. Zhu, Asymptotic-preserving methods for hyperbolic and transport equations with random input and diffusive scalings. J. Comput. Phys. 289, 35–52 (2015)

    Article  MathSciNet  Google Scholar 

  24. S. Jin, J.G. Liu, 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 (2016)

    Google Scholar 

  25. Q. Li, L. Wang, Uniform regularity for linear kinetic equations with random input based on hypocoercivity (2016). arXiv:1612.01219

  26. Q. Li, J. Lu, W. Sun, A convergent method for linear half-space kinetic equation. Math. Model. Numer. Anal. (in press)

    Google Scholar 

  27. Q. Li, J. Lu, W. Sun, Half-space kinetic equations with general boundary conditions. Math. Comput. (in press)

    Google Scholar 

  28. Q. Li, J. Lu, W. Sun, Validity and regularization of classical half-space equations. J. Stat. Phys. (in press)

    Google Scholar 

  29. Q. Li, J. Lu, W. Sun, Diffusion approximations of linear transport equations: asymptotics and numerics. J. Comput. Phys. 292, 141–167 (2015)

    Article  MathSciNet  Google Scholar 

  30. F. Nobile, R. Tempone, C. Webster, A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 46(3), 2309–2345 (2008)

    Article  MathSciNet  Google Scholar 

  31. F. Nobile, R. Tempone, C.G. Webster, An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 46(5), 2411–2442 (2008)

    Article  MathSciNet  Google Scholar 

  32. F. Nobile, R. Tempone, C.G. Webster, A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 46(5), 2309–2345 (2008)

    Article  MathSciNet  Google Scholar 

  33. C. Schwab, R.-A. Todor, Sparse finite elements for elliptic problems with stochastic loading. Numer. Math. 95(4), 707–734 (2003)

    Article  MathSciNet  Google Scholar 

  34. D. Xiu, J.S. Hesthaven, High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27(3), 1118–1139 (2005)

    Article  MathSciNet  Google Scholar 

  35. D. Xiu, G.E. Karniadakis, Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phys. 187(1), 137–167 (2003)

    Article  MathSciNet  Google Scholar 

  36. G. Zhang, M. Gunzburger, Error analysis of a stochastic collocation method for parabolic partial differential equations with random input data. SIAM J. Numer. Anal. 50(4), 1922–1940 (2012)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics at Würzburg University, Emil Fischer Str. 40, 97074, Würzburg, Germany

    Christian Klingenberg & Qin Li

  2. University of Wisconsin Madison, 480 Lincoln Dr., Madison, WI, 53705, USA

    Marlies Pirner

Authors
  1. Christian Klingenberg
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qin Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Marlies Pirner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christian Klingenberg .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Würzburg University, Würzburg, Germany

    Christian Klingenberg

  2. Department of Mathematics, RWTH Aachen University, Aachen, Germany

    Michael Westdickenberg

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Klingenberg, C., Li, Q., Pirner, M. (2018). On Quantifying Uncertainties for the Linearized BGK Kinetic Equation. In: Klingenberg, C., Westdickenberg, M. (eds) Theory, Numerics and Applications of Hyperbolic Problems II. HYP 2016. Springer Proceedings in Mathematics & Statistics, vol 237. Springer, Cham. https://doi.org/10.1007/978-3-319-91548-7_14

Download citation

Publish with us

Policies and ethics

Search

Navigation