Galerkin Methods for Stochastic Hyperbolic Problems Using Bi-Orthogonal Polynomials

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.

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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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Fishman, G.: Monte Carlo: Concepts, Algorithms, and Applications. Springer, New York (1996)

    Google Scholar 

  4. 4.

    Foo, J., Karniadakis, G.E.: Multi-element probabilistic collocation method in high dimensions. J. Comput. Phys. 229, 1536–1557 (2010)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Ghanem, R.G., Spanos, P.: Stochastic Finite Elements: A Spectral Approach. Springer, Berlin (1991)

    Google Scholar 

  8. 8.

    Gottlieb, D., Xiu, D.: Galerkin method for wave equations with uncertain coefficients. Commun. Comput. Phys. 3, 505–518 (2008)

    MathSciNet  MATH  Google 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)

    MathSciNet  Article  Google 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)

    MATH  Article  Google Scholar 

  12. 12.

    Liu, W., Belytschko, T., Mani, A.: Random field finite elements. Int. J. Numer. Methods Eng. 23, 1831–1845 (1986)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  Google 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)

    MathSciNet  MATH  Article  Google 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)

    MATH  Article  Google Scholar 

  16. 16.

    Wiener, N.: The homogeneous chaos. Am. J. Math. 60, 897–936 (1938)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Xiu, D.: Fast numerical methods for stochastic computations: A review. Commun. Comput. Phys. 5, 242–272 (2009)

    MathSciNet  Google Scholar 

  18. 18.

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

    MathSciNet  MATH  Article  Google 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

  20. 20.

    Xiu, D., Shen, J.: Efficient stochastic Galerkin methods for random diffusion equations. J. Comput. Phys. 228, 266–281 (2009)

    MathSciNet  MATH  Article  Google Scholar 

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Correspondence to Tao Tang.

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Zhou, T., Tang, T. Galerkin Methods for Stochastic Hyperbolic Problems Using Bi-Orthogonal Polynomials. J Sci Comput 51, 274–292 (2012). https://doi.org/10.1007/s10915-011-9508-0

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Keywords

  • Hyperbolic equation
  • Regularity
  • Stochastic Galerkin methods
  • Bi-orthogonal