Galerkin Methods for Stochastic Hyperbolic Problems Using Bi-Orthogonal Polynomials
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.
KeywordsHyperbolic equation Regularity Stochastic Galerkin methods Bi-orthogonal
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- 5.Fox, B.: Strategies for Quasi-Monte Carlo. Kluwer Academic, Dordrecht (1999) Google Scholar
- 9.Golub, G., van Loan, C.: Matrix Computation, 3rd edn. The John Hopkins University Press, Baltimore (1996) Google Scholar
- 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