Computation of statistical solutions of hyperbolic systems of conservation laws

Abstract

Statistical solutions are time-parameterized probability measures on spaces of integrable functions, that have been proposed recently as a framework for global solutions and uncertainty quantification for multi-dimensional hyperbolic system of conservation laws. By combining high-resolution finite volume methods with a Monte Carlo sampling procedure, we present a numerical algorithm to approximate statistical solutions. Under verifiable assumptions on the finite volume method, we prove that the approximations, generated by the proposed algorithm, converge in an appropriate topology to a statistical solution. Numerical experiments illustrating the convergence theory and revealing interesting properties of statistical solutions, are also presented. We furthermore show that the multi-level Monte Carlo algorithm converges in the weak topology, and provide testable conditions for when the multi-level Monte Carlo algorithm outperforms the Monte Carlo algorithm. Finally we present the Alsvinn simulator, a fast multi general purpose graphical processing unit (GPGPU) finite volume solver for hyperbolic conservation laws in multiple space dimensions. Alsvinn has native support for uncertainty quantifications, and exhibits excellent scaling on top tier compute clusters.