Uncertainty quantification for hyperbolic conservation laws with flux coefficients given by spatiotemporal random fields

TL;DR

This paper develops a Monte Carlo Finite Volume method to quantify uncertainty in hyperbolic PDEs with flux coefficients modeled by spatiotemporal Gaussian and Ornstein-Uhlenbeck random fields, demonstrating its effectiveness on physical models.

Contribution

It introduces a novel discretization framework for stochastic hyperbolic PDEs with complex spatiotemporal random coefficients, including error analysis and convergence results.

Findings

Successful computation of solution moments for random hyperbolic PDEs

Application to magnetic induction and acoustics with spatiotemporal randomness

Robust numerical method validated on physical models

Abstract

In this paper hyperbolic partial differential equations with random coefficients are discussed. We consider the challenging problem of flux functions with coefficients modeled by spatiotemporal random fields. Those fields are given by correlated Gaussian random fields in space and Ornstein-Uhlenbeck processes in time. The resulting system of equations consists of a stochastic differential equation for each random parameter coupled to the hyperbolic conservation law. We define an appropriate solution concept in his setting and analyze errors and convergence of discretization methods. A novel discretization framework, based on Monte Carlo Finite Volume methods, is presented for the robust computation of moments of solutions to those random hyperbolic partial differential equations. We showcase the approach on two examples which appear in applications: The magnetic induction equation and linear acoustics, both with a spatiotemporal random background velocity field.