Solving Random Hyperbolic Conservation Laws Using Linear Programming

SIAM Journal on Scientific Computing, vol. 48, pp. A1184–A1205

Abstract

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 for the computation of 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.

Authors 4

1. Shaoshuai Chu Aachen Institut für Geometrie und Praktische Mathematik

   RWTH Aachen University

   Affiliation as printed

   Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, 52056 Aachen, Germany

2. Michaël Herty Aachen Institut für Geometrie und Praktische Mathematik

   RWTH Aachen University

   Affiliation as printed

   Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, 52056 Aachen, Germany

3. Mária Lukáčová-Medviďová

   Johannes Gutenberg University Mainz

   Affiliation as printed

   Institute of Mathematics, Johannes Gutenberg University Mainz, Staudingerweg 9, 55128 Mainz, Germany

4. Yizhou Zhou Aachen Institut für Geometrie und Praktische Mathematik

   RWTH Aachen University

   Affiliation as printed

   Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, 52056 Aachen, Germany