Robust Uncertainty Propagation in Systems of Conservation Laws with the Entropy Closure Method

Abstract

In this paper, we consider hyperbolic systems of conservation laws subject to uncertainties in the initial conditions and model parameters. In order to solve the underlying uncertain systems, we rely on moment theory and the construction of a moment model in the framework of parametric polynomial approximations. We prove the spectral convergence of this approach for the uncertain inviscid Burgers’ equation. We also emphasize the difficulties arising when applying the standard moment method in the context of uncertain systems of conservation laws. In particular, we focus on two relevant examples: the shallow water equations and the Euler system. Next, we review the entropy-based method inspired by plasma physics and rational extended thermodynamics that we propose in this context. We then study the mathematical structure of the well-posed large systems of discretized partial differential equations arising in this framework. The first aim of this work is the description of some mathematical features of the moment method applied to the modeling of uncertainties in systems of conservation laws. The second objective is to relate theoretical description and understanding to some basic numerical results obtained for the numerical approximation of such uncertain models. All numerical examples come from fluid dynamics inspired problems.