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Volume 3, Issue 2
Galerkin Method for Wave Equations with Uncertain Coefficients

David Gottlieb & Dongbin Xiu

Commun. Comput. Phys., 3 (2008), pp. 505-518.

Published online: 2008-03

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  • Abstract

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.

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@Article{CiCP-3-505, author = {David Gottlieb and Dongbin Xiu}, title = {Galerkin Method for Wave Equations with Uncertain Coefficients}, journal = {Communications in Computational Physics}, year = {2008}, volume = {3}, number = {2}, pages = {505--518}, abstract = {

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.

}, issn = {1991-7120}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/cicp/7864.html} }
TY - JOUR T1 - Galerkin Method for Wave Equations with Uncertain Coefficients AU - David Gottlieb & Dongbin Xiu JO - Communications in Computational Physics VL - 2 SP - 505 EP - 518 PY - 2008 DA - 2008/03 SN - 3 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/cicp/7864.html KW - AB -

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.

David Gottlieb and Dongbin Xiu. (2008). Galerkin Method for Wave Equations with Uncertain Coefficients. Communications in Computational Physics. 3 (2). 505-518. doi:
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