Convergence Analysis for Stochastic Collocation Methods to Scalar Hyperbolic Equations with a Random Wave Speed
Tao Tang 1*, Tao Zhou 21 Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong.
2 Institute of Computational Mathematics, Academy of Mathematics and System Science, Chinese Academy of Sciences, Beijing 100190, China.
Received 6 January 2009; Accepted (in revised version) 13 January 2010
Available online 23 February 2010
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].
AMS subject classifications: 52B10, 65D18, 68U05, 68U07
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Key words: Hyperbolic equation, stochastic collocation methods, convergence analysis.
Email: email@example.com (T. Tang), firstname.lastname@example.org (T. Zhou)