Convergence Analysis for Stochastic Collocation Methods to Scalar Hyperbolic Equations with a Random Wave Speed (Communications in Computational Physics, 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 generalizedpolynomial chaos (gPC) method and demonstrated that when uncertaintycauses the change of characteristic directions, the resulting deterministic system ofequations is a symmetric hyperbolic system with both positive and negative eigenvalues.Consequently, a consistent method of imposing the boundary conditions isproposed and its convergence is established under the assumption that the expansioncoefficients decay fast asymptotically. In this work, we investigate stochastic collocationmethods for the same type of scalar wave equation with random wave speed. Itwill 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 andboundary data. Numerical examples are presented to support the analysis and also toshow the sharpness of the assumptions on the relationship between the random wavespeed and the initial and boundary data. An accuracy enhancement technique is investigatedfollowing the multi-element collocation method proposed by Foo, Wan andKarniadakis [J. Comput. Phys., 227 (2008), pp. 9572-9595].