Method of Distributions for Systems with Stochastic Forcing

TL;DR

This paper develops a distribution-based method for systems with stochastic forcing governed by hyperbolic conservation laws, providing a deterministic equation for the CDF without closure approximations, verified against Monte Carlo simulations.

Contribution

It introduces a novel distribution method for stochastic hyperbolic systems that directly yields the CDF, avoiding closure approximations and validated through numerical comparisons.

Findings

Accurately predicts mean and standard deviation for Gaussian, normal, and beta distributions.

Matches Monte Carlo simulation results closely.

Provides a deterministic CDF equation for stochastic conservation laws.

Abstract

The method of distributions is developed for systems that are governed by hyperbolic conservation laws with stochastic forcing. The method yields a deterministic equation for the cumulative density distribution (CDF) of a system state, e.g., for flow velocity governed by an inviscid Burgers' equation with random source coefficients. This is achieved without recourse to any closure approximation. The CDF model is verified against MC simulations using spectral numerical approximations. It is shown that the CDF model accurately predicts the mean and standard deviation for Gaussian, normal and beta distributions of the random coefficients.