Analysis and Computation of Second Order Stochastic Hyperbolic Equations

Abstract eng:
Following the works on stochastic elliptic and parabolic problems [1,2], we consider the problem of numerically approximating statistical moments of the solution of linear second order hyperbolic partial differential equations with random coefficients and deterministic initial and boundary data. The coefficients are assumed to depend on a finite number of random variables. In particular, we study the wave field in multi-layered media, where the number of layers determines the number of random variables. We show that the problem is well-posed under suitable assumptions on the data. We propose and analyze a stochastic collocation method to numerically solve the problem. In this method, the problem is first discretized in space and time using a deterministic numerical method, such as the finite element or finite difference method, to obtain a semi-discrete problem. Next, the semi-discrete problem is collocated in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space. This leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. We consider both full and sparse tensor product spaces of orthogonal polynomials. We provide a rigorous convergence analysis and demonstrate different types of convergence of the “probability error” with respect to the number of collocation points for different tensor product spaces and under some regularity assumptions on the data. In particular, we show that, unlike in elliptic and parabolic problems, the solution to hyperbolic problems is not in general analytic with respect to the random variables. Therefore, in general, the rate of convergence is only algebraic. An exponential rate of convergence is still possible for particular types of data. We present numerical examples, which confirm the analysis and show that the collocation method is a valid alternative to the more traditional Monte Carlo method for this class of problems.

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