An Efficient Solver for Cumulative Density Function-based Solutions of Uncertain Kinematic Wave Models

TL;DR

The paper addresses forward uncertainty quantification in kinematic wave models by leveraging a fine-grained cumulative density function (CDF) framework. It develops an efficient solver that reduces the stochastic problem to solving a linear Pi equation along characteristics, requiring only ensemble averages of Pi to recover the system's CDF F_k. Through 1D, 3D, coupled, and Burgers' equation tests, the method demonstrates high accuracy and substantial speedups over direct Monte Carlo simulations, with rapid convergence in the number of realizations. The approach enables reliable probabilistic predictions for environmental flows (e.g., open-channel and flood waves) with lower computational cost, making it attractive for risk assessment and decision support in hydrology and hydraulics.

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 kinematic wave model. The approach relies on Monte Carlo Simulations (MCS) of the fine-grained CDF equation of system state, as derived by the CDF method. This fine-grained CDF equation is solved via the method of characteristics. Each method of characteristics solution is far more computationally efficient than the direct solution of the kinematic wave model, and the MCS estimator of the CDF converges relatively quickly. We verify the accuracy and robustness of our procedure via comparison with direct MCS of a particular kinematic wave system, the Saint-Venant equation.

Figures (6)

• Figure 1: (a) Cumulative density function $F_{\mathrm k}(K; x=0.2, \, t=1)$ from the proposed CDF method $(CDF)$ and the exact $(exact)$ solution \ref{['sol:test2-CDF']}; (b) their relative error $\epsilon$\ref{['error:1D-random']} for different number of realizations.
• Figure 2: $l_2$-norm error \ref{['error:1D-random']} of the CDF solutions $F_{\mathrm k}(K; x_1=1.3, x_2=1.3, x_3=1.3 \, t=1)$ between the exact solution \ref{['sol:3d']} and the one obtained from the proposed CDF scheme, and that from the Monte Carlo Simulations (MCS) of the original system \ref{['eq:3d']}, respectively, at different realization number $M$.
• Figure 3: The $l_2$-norm errors \ref{['error:1D-random']} of a) $F_{\mathrm k_1}(k_1'; x=0.3 , t=1)$ and b) $F_{\mathrm k_2}(k_2'; x=0.3 , t=1)$, between the exact solutions \ref{['sol:couple']} and those from the CDF method, and MCS of the original coupled system, respectively.
• Figure 4: The $l_2$-norm errors \ref{['error:1D-random']} between the converged solution $F_{\mathrm k}(K; x=0.4, \, t=1)$ and those from CDF method (CDF), and those from Monte Carlo simulations (MCS) of the Burger's equation, respectively, at different realization numbers $M$. The converged solution is obtained from $2000$ MCS simulations.
• Figure 5: Cumulative density function of flux $F_{\mathrm q}(Q; x=1, \, t=1)$ from CDF scheme (CDF) and Monte Carlo Simulations (MCS) of the kinematic wave equation \ref{['kwt-saint-venant']}, under $S=0$, $S=1$ and $S=x$ conditions.