Galerkin Method for Wave Equations with Uncertain Coefficients

By David I. Gottlieb, Dongbin Xiu. Communications in Computational Physics, 2008

148 citations recorded by the source corpus.

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

Subfields: Civil and Structural Engineering, Environmental Engineering, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/e91183b70c58d49aff8408183d8176c98fad50b5

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Civil and Structural Engineering

In Environmental Engineering

In Statistics and Probability

Citations in the corpus

Fast Numerical Methods for Stochastic Computations: A Review (2009, background): “The convergence of gPC for such functions was discussed in [31, 81].”
Convergence Analysis for Stochastic Collocation Methods to Scalar Hyperbolic Equations with a Random Wave Speed (2010, methodology): “Gottlieb and Xiu [7] made the first attempt by considering a simple model of a scalar wave equation with random wave speeds.”