An efficient solver for cumulative density function-based solutions of uncertain kinematic wave models

Abstract

We develop a numerical framework to implement the cumulative density function (CDF) method for obtaining the probability distribution of the system state described by a stochastic kinematic wave model. The approach relies on the computation of the fine-grained CDF equation of system state, as derived by the CDF method, via the method of characteristics. Due to its linearity, the fine-grained CDF equation is solved far more efficiently than the direct simulation of the kinematic wave model. Since the ensemble mean of the fine-grained CDF solutions is the probability distribution of the original system state, the proposed scheme requires less realizations than direct Monte Carlo simulations (MCS) of the kinematic model and thus converges relatively quickly. We verify the accuracy and effectiveness of our procedure via comparisons with direct MCS of several examples, including a particular kinematic wave system, the Saint-Venant equation.