Solving Random Hyperbolic Conservation Laws Using Linear Programming

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.