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

## TL;DR

This paper introduces a computationally efficient numerical framework using the CDF method and Monte Carlo simulations to determine the probability distribution of uncertain kinematic wave models, validated against the Saint-Venant equation.

## Contribution

The paper presents a novel, efficient solver for the CDF-based approach to uncertain kinematic wave models using the method of characteristics and Monte Carlo simulations.

## Key findings

- The proposed method is significantly faster than direct Monte Carlo simulations.
- The solver accurately reproduces the probability distribution of the system state.
- Validation against the Saint-Venant equation confirms robustness and accuracy.

## 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.