Convergence analysis for stochastic collocation methods to scalar hyperbolic equations with a random wave speed

Tao TANG*, Tao Zhou

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

33 Citations (Scopus)

Abstract

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].

Original languageEnglish
Pages (from-to)226-248
Number of pages23
JournalCommunications in Computational Physics
Volume8
Issue number1
DOIs
Publication statusPublished - Jul 2010

Scopus Subject Areas

  • Physics and Astronomy (miscellaneous)

User-Defined Keywords

  • Convergence analysis
  • Hyperbolic equation
  • Stochastic collocation methods

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