# KAUST-IAMCS Workshop on Multiscale Modeling, Advanced Discretization Techniques, and Simulation of Wave Propagation

## Raul Tempone, King Abdullah University of Science and Technology Stochastic Collocation for Second Order Hyperbolic Equations with Random Coefficients

### Authors

• Mohammad Motamed, King Abdullah University of Science and Technology
• Fabio Nobile, Swiss Federal Institute of Technology in Lausanne (Switzerland) and Polytechnic University of Milan (Italy)
• Raul Tempone, King Abdullah University of Science and Technology

### Abstract

We propose and analyze a stochastic collocation method for solving the second order wave equation with a random wave speed and deterministic boundary and initial conditions. Here, the speed depends 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. Moreover, we provide a rigorous convergence analysis and demonstrate different types of convergence of the probability error with respect to the number of collocation points. 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 input random variables. Therefore, in general, the rate of convergence in the approximation of the solution is only algebraic. A rate of convergence faster than algebraic is still possible for particular types of input data and or quantities of interest. 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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