@Article{CiCP-8-226,
author = {Tao Tang and Tao Zhou},
title = {Convergence Analysis for Stochastic Collocation Methods to Scalar Hyperbolic Equations with a Random Wave Speed},
journal = {Communications in Computational Physics},
year = {2010},
volume = {8},
number = {1},
pages = {226--248},
abstract = {<p style="text-align: justify;">For a simple model of a scalar wave equation with a random wave speed,
Gottlieb and Xiu [Commun. Comput. Phys., 3 (2008), pp. 505-518] employed the generalized
polynomial chaos (gPC) method and demonstrated 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.
Consequently, a consistent method of imposing the boundary conditions is
proposed and its convergence is established under the assumption that the expansion
coefficients decay fast asymptotically. In this work, we investigate stochastic collocation
methods for the same type of scalar wave equation with random wave speed. It
will be demonstrated that the rate of convergence depends on the regularity of the solutions;
and the regularity is determined by the random wave speed and the initial and
boundary data. Numerical examples are presented to support the analysis and also to
show the sharpness of the assumptions on the relationship between the random wave
speed and the initial and boundary data. An accuracy enhancement technique is investigated
following the multi-element collocation method proposed by Foo, Wan and
Karniadakis [J. Comput. Phys., 227 (2008), pp. 9572-9595].</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.060109.130110a},
url = {http://global-sci.org/intro/article_detail/cicp/7570.html}
}