Solving Wick-stochastic water waves using a Galerkin finite element method

Abstract

A Galerkin finite element approximation of Wick-stochastic water waves is developed and numerically investigated. The problems under study consist of a class of shallow water equations driven by white noise. Random effects may appear in the water free surface or in the bottom topography among others. To perform a rigorous study of stochastic effects in the shallow water equations we employ techniques from Wick calculus. The differentiation respect to time and space along with the product operations are performed in a distribution sense. Using the Wiener-Itô chaos expansion for treating the randomness, the governing equations are transformed into a sequence of deterministic shallow water equations to be solved for each chaos coefficient by standard methods from computational fluid dynamics. In our study, we formulate a finite element method for spatial discretization and a backward Euler scheme for time integration. Once the chaos coefficients are obtained, statistical moments for the stochastic solution are carried out. Numerical results are presented for stochastic water waves in the Strait of Gibraltar.