Uncertainty analysis of simplified 1D and 2D shallow water equations via the Karhunen–Loéve expansion and Monte Carlo simulations

Abstract

The stochastic solution of wave propagation through simplified shallow water equations, described by a system of 1D and 2D linear equations, has been investigated by considering the initial condition as a source of uncertainty. The Karhunen–Loéve expansion (KLE) method is applied as an alternative approach to the Monte Carlo simulation (MCS) method. Uncertainty associated with moments of the flow characteristics is quantified. The initial condition ${H}_{0}$, considered as the input random field, is decomposed in the form of a set of orthogonal Gaussian random variables $\left\{{\xi }_{i}\right\}$. The coefficients of the series are related to eigenvalues and eigenfunctions of the covariance function of ${H}_{0}$. The flow depth $H$ and flow velocities $U$ and V are expanded as an infinite series whose terms H⁽ⁿ⁾, U⁽ⁿ⁾ and V⁽ⁿ⁾ represent depth and velocities of the nth order, respectively, and a set of recursive equations is derived for H⁽ⁿ⁾, U⁽ⁿ⁾ and V⁽ⁿ⁾. Then, H⁽ⁿ⁾, U⁽ⁿ⁾ and V⁽ⁿ⁾ are decomposed with polynomial expansions in terms of the products of ${\xi }_{i}$ so that their coefficients are determined by replacing decompositions of H₀, H⁽ⁿ⁾, U⁽ⁿ⁾ and V⁽ⁿ⁾ into those recursive equations. MCS is conducted, in which the mean and variances of the flow depth $H$ and flow velocities $U$ and V are compared against approximations of the KLE results, with the same accuracy as MCS, yet with much less computation time and effort.