KAUST-IAMCS Workshop on Multiscale Modeling, Advanced Discretization Techniques, and Simulation of Wave Propagation
- King Abdullah University of Science and Technology (KAUST)
- Thuwal, Saudi Arabia
- Mohammad Motamed, King Abdullah University of Science and Technology
- A Collocation Method for Wave Equation with Random Coefficients
We propose and analyze a stochastic collocation method for solving the second order wave equation with a random wave speed and subjected to deterministic boundary and initial conditions. The speed is assumed to depend on the physical space and a finite number of random variables. The numerical scheme consists of a finite difference or finite element method in the physical space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space. This approach leads to the solution of uncoupled deterministic problems as in the Monte Carlo method. 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.