High-Order Perturbation Approach for Wave Transformation by Applying Advection-Diffusion Equation via Karhunen–Loeve Expansion

Abstract

An efficient approach, termed the Karhunen–Loeve expansion (KLE), for uncertainty analysis of flow in open-channel flow is applied. The initial condition, as a random field, is described in the form of two consecutive solitary waves that are propagated through the advection-diffusion equation (ADE). The aim of this paper is to quantify the uncertainty associated with flow depth moments such as mean flow depth and flow depth variance. In the proposed approach, the initial condition, h(x), is decomposed as an infinite series on the basis of a set of orthogonal Gaussian standard random variables. Eigenvalues and eigenfunctions of the covariance function of initial condition, which are extracted from Fredhulm’s equation, play a key role in the coefficients of the series. Then, flow depth H(x,t) is written as an infinite series, where every term of H(m) corresponds to flow depth of the mth order in terms of standard deviation of the input random field. H(m) is decomposed in terms of the products of m Gaussian random variables and unknown coefficients are determined by solving the ADE recursively when h and H(m) were substituted. To validate the proposed approach, the resulting mean and variance of the flow quantities are compared to those from Monte Carlo simulations (MCS) as a reliable method. It is found that the proposed approach can accurately approximate the flow depth statistics in a more computationally efficient manner than the MCS.