Solving Random Hyperbolic Conservation Laws Using Linear Programming

TL;DR

This paper introduces a new linear programming-based numerical method for solving random hyperbolic conservation laws, leveraging measure-valued solutions and Young measures to preserve structure and handle stochasticity.

Contribution

The paper presents a novel measure-valued solution approach using linear programming for random hyperbolic conservation laws, enabling structure preservation and stochastic analysis.

Findings

Demonstrated effectiveness on Burgers and Euler equations

Compared favorably with stochastic collocation methods

Showed how entropy choices affect solution measures

Abstract

A novel structure-preserving numerical method to solve random hyperbolic systems of conservation laws is presented. The method uses a concept of generalized, measure-valued solutions to random conservation laws. This yields a linear partial differential equation with respect to the Young measure and allows to compute the approximation based on linear programming problems. We analyze structure-preserving properties of the derived numerical method and discuss its advantages and disadvantages. We numerically demonstrate the approach on the one-dimensional Burgers and isentropic Euler equations and compare with stochastic collocation. In addition, we introduce a discontinuous-flux test in which different entropies used in the linear-program objective select different weak entropy solutions, and we report the corresponding changes in the moments and supports of the Young measure.