Journal of Scientific Computing

, Volume 51, Issue 2, pp 274–292 | Cite as

Galerkin Methods for Stochastic Hyperbolic Problems Using Bi-Orthogonal Polynomials

Article

Abstract

This work is concerned with scalar transport equations with random transport velocity. We first give some sufficient conditions that can guarantee the solution to be in appropriate random spaces. Then a Galerkin method using bi-orthogonal polynomials is proposed, which decouples the equation in the random spaces, yielding a sequence of uncoupled equations. Under the assumption that the random wave field has a structure of the truncated KL expansion, a principle on how to choose the orders of the approximated polynomial spaces is given based on the sensitivity analysis in the random spaces. By doing this, the total degree of freedom can be reduced significantly. Numerical experiments are carried out to illustrate the efficiency of the proposed method.

Keywords

Hyperbolic equation Regularity Stochastic Galerkin methods Bi-orthogonal 

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References

  1. 1.
    Babuška, I., Tempone, R., Zouraris, G.: Galerkin finite element approximations of stochastic elliptic differential equations. SIAM J. Numer. Anal. 42, 800–825 (2004) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Babuška, I., Jobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45, 1005–1034 (2007) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Fishman, G.: Monte Carlo: Concepts, Algorithms, and Applications. Springer, New York (1996) MATHGoogle Scholar
  4. 4.
    Foo, J., Karniadakis, G.E.: Multi-element probabilistic collocation method in high dimensions. J. Comput. Phys. 229, 1536–1557 (2010) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Fox, B.: Strategies for Quasi-Monte Carlo. Kluwer Academic, Dordrecht (1999) Google Scholar
  6. 6.
    Gerstner, T., Griebel, M.: Dimension-adaptive tensor-product quadrature. Computing 71(1), 65–87 (2003) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Ghanem, R.G., Spanos, P.: Stochastic Finite Elements: A Spectral Approach. Springer, Berlin (1991) MATHCrossRefGoogle Scholar
  8. 8.
    Gottlieb, D., Xiu, D.: Galerkin method for wave equations with uncertain coefficients. Commun. Comput. Phys. 3, 505–518 (2008) MathSciNetMATHGoogle Scholar
  9. 9.
    Golub, G., van Loan, C.: Matrix Computation, 3rd edn. The John Hopkins University Press, Baltimore (1996) Google Scholar
  10. 10.
    Jin, C., Cai, X.: A preconditioned recycling GMRES solver for stochastic Helmholtz problems. Commun. Comput. Phys. 6, 342–353 (2009) MathSciNetCrossRefGoogle Scholar
  11. 11.
    Liu, W., Belytschko, T., Mani, A.: Probabilistic finite elements for nonlinear structural dynamics. Comput. Methods Appl. Mech. Eng. 56, 61–81 (1986) MATHCrossRefGoogle Scholar
  12. 12.
    Liu, W., Belytschko, T., Mani, A.: Random field finite elements. Int. J. Numer. Methods Eng. 23, 1831–1845 (1986) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Tang, T., Zhou, T.: Convergence analysis for stochastic collocation methods to scalar hyperbolic equations with a random wave speed. Commun. Comput. Phys. 8, 226–248 (2010) MathSciNetGoogle Scholar
  14. 14.
    Wan, X., Karniadakis, G.E.: Long-term behavior of polynomial chaos in stochastic flow simulations. Comput. Methods Appl. Mech. Eng. 195, 5582–5596 (2006) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Werder, T., Gerdes, K., Schötzau, D., Schwab, C.: hp-discontinuous Galerkin time stepping for parabolic problems. Comput. Methods Appl. Mech. Eng. 190, 6685–6708 (2001) MATHCrossRefGoogle Scholar
  16. 16.
    Wiener, N.: The homogeneous chaos. Am. J. Math. 60, 897–936 (1938) MathSciNetCrossRefGoogle Scholar
  17. 17.
    Xiu, D.: Fast numerical methods for stochastic computations: A review. Commun. Comput. Phys. 5, 242–272 (2009) MathSciNetGoogle Scholar
  18. 18.
    Xiu, D., Karniadakis, G.E.: Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phys. 187, 137–167 (2003) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Xiu, D., Karniadakis, G.E.: The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 Google Scholar
  20. 20.
    Xiu, D., Shen, J.: Efficient stochastic Galerkin methods for random diffusion equations. J. Comput. Phys. 228, 266–281 (2009) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Institute of Computational Mathematics, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  2. 2.Department of MathematicsHong Kong Baptist UniversityKowloon TongChina

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