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Galerkin Methods for Stochastic Hyperbolic Problems Using Bi-Orthogonal Polynomials

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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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Authors and Affiliations

  1. Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Tao Zhou

  2. Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China

    Tao Tang

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  1. Tao Zhou
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  2. Tao Tang
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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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  • DOI: https://doi.org/10.1007/s10915-011-9508-0

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