In this talk I will first review the stochastic Galerkin method for the propagation of input-data uncertainty in partial differential equations. The case of models, with random parameters, expressed by systems of stochastic conservation laws will then be considered. An analysis of the deterministic conservative system resulting from the Galerkin projection of the stochastic model will be provided, emphasizing on its hyperbolic character. Based on these theoretical results, a Roe-type solver is presented and an algorithm for the fast approximation of the upwinding matrix is proposed.
An essential characteristic of hyperbolic model solutions is the presence of singularities and discontinuities having random velocities. To deal with non-smooth solutions we introduce a multi-resolution framework for the stochastic discretization. In this framework, the stochastic discretization is adapted in space and time to follow the local smoothness of the solution.
Examples of simulation will be shown for the Burger's and traffic equations (scalar problem), and Euler system (compressible flows).
Olivier Le Maître is researcher at the Centre National de la Recherche Scientifique (CNRS), France, since 2007. He graduated in mechanical engineering and aero-thermo-chemistry at the Université du Havre in 1995; PhD in computational mechanics at the Université du Havre in 1998; Habilitation Diploma at the Université d'Evry in 2006.
From 1998 to 2007, he was Assistant Professor at the mechanical engineering department of the Université d'Evry, Paris. In 2007, he was appointed as researcher by the CNRS and joined the LIMSI (Computer and Engineering Sciences Laboratory) in Orsay, where he was promoted Research Director in 2011. Since 2009, he is deputy director of the National Research Group on modeling and mathematics for nuclear waste disposal simulations.
He was invited multiple times to the Johns Hopkins and Duke universities and Sandia National Labs as a visiting researcher or professor. He is author of about 45 publications on international journals. His main research activities concern uncertainty quantification methods, computational fluid dynamics, numerical approximation of PDEs and reactive systems.