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.

[1]  George E. Karniadakis,et al.  Multi-element probabilistic collocation method in high dimensions , 2010, J. Comput. Phys..

[2]  Raúl Tempone,et al.  Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations , 2004, SIAM J. Numer. Anal..

[3]  Wing Kam Liu,et al.  Random field finite elements , 1986 .

[4]  B. Fox.  Strategies for Quasi-Monte Carlo , 1999, International Series in Operations Research & Management Science.

[5]  Ronald L. Wasserstein,et al.  Monte Carlo: Concepts, Algorithms, and Applications , 1997 .

[6]  Fabio Nobile,et al.  A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data , 2008, SIAM J. Numer. Anal..

[7]  Tao Tang,et al.  Convergence Analysis for Stochastic Collocation Methods to Scalar Hyperbolic Equations with a Random Wave Speed , 2010 .

[8]  D. Xiu,et al.  Modeling uncertainty in flow simulations via generalized polynomial chaos , 2003 .

[9]  Hermann G. Matthies,et al.  Numerical Methods and Smolyak Quadrature for Nonlinear Stochastic Partial Differential Equations , 2003 .

[10]  G. Karniadakis,et al.  Long-Term Behavior of Polynomial Chaos in Stochastic Flow Simulations , 2006 .

[11]  D. Xiu.  Fast numerical methods for stochastic computations: A review , 2009 .

[12]  BabuskaIvo,et al.  A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data , 2007 .

[13]  Xiao-Chuan Cai,et al.  A Preconditioned Recycling GMRES Solver for Stochastic Helmholtz Problems , 2009 .

[14]  R. Ghanem,et al.  Stochastic Finite Elements: A Spectral Approach , 1990 .

[15]  Thomas Gerstner,et al.  Dimension–Adaptive Tensor–Product Quadrature , 2003, Computing.

[16]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[17]  Wing Kam Liu,et al.  Probabilistic finite elements for nonlinear structural dynamics , 1986 .

[18]  N. Cutland,et al.  On homogeneous chaos , 1991, Mathematical Proceedings of the Cambridge Philosophical Society.

[19]  Gene H. Golub,et al.  Matrix computations , 1983 .

[20]  Jie Shen,et al.  Efficient stochastic Galerkin methods for random diffusion equations , 2009, J. Comput. Phys..

[21]  Dongbin Xiu,et al.  Galerkin method for wave equations with uncertain coefficients , 2008 .

[22]  Fabio Nobile,et al.  A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data , 2007, SIAM Rev..

[23]  Dominik Schötzau,et al.  hp-discontinuous Galerkin time stepping for parabolic problems , 2001 .