Commun. Comput. Phys.,
Galerkin Method for Wave Equations with Uncertain Coefficients
David Gottlieb 1, Dongbin Xiu 2*1 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA.
2 Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA.
Received 12 April 2007; Accepted (in revised version) 2 October 2007
Available online 25 October 2007
Polynomial chaos methods (and generalized polynomial chaos methods) have been extensively applied to analyze PDEs that contain uncertainties. However this approach is rarely applied to hyperbolic systems. In this paper we analyze the properties of the resulting deterministic system of equations obtained by stochastic Galerkin projection. We consider a simple model of a scalar wave equation with random wave speed. We show that when uncertainty causes the change of characteristic directions, the resulting deterministic system of equations is a symmetric hyperbolic system with both positive and negative eigenvalues. A consistent method of imposing the boundary conditions is proposed and its convergence is established. Numerical examples are presented to support the analysis.AMS subject classifications: 65C20, 65C30
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Key words: Generalized polynomial chaos, stochastic PDE, Galerkin method, hyperbolic equation, uncertainty quantification.
Email: firstname.lastname@example.org (D. Gottlieb), email@example.com (D. Xiu)