a b s t r a c tThis paper deals with stochastic spectral methods for uncertainty propagation and quantification in nonlinear hyperbolic systems of conservation laws. We consider problems with parametric uncertainty in initial conditions and model coefficients, whose solutions exhibit discontinuities in the spatial as well as in the stochastic variables. The stochastic spectral method relies on multi-resolution schemes where the stochastic domain is discretized using tensor-product stochastic elements supporting local polynomial bases. A Galerkin projection is used to derive a system of deterministic equations for the stochastic modes of the solution. Hyperbolicity of the resulting Galerkin system is analyzed. A finite volume scheme with a Roe-type solver is used for discretization of the spatial and time variables. An original technique is introduced for the fast evaluation of approximate upwind matrices, which is particularly well adapted to local polynomial bases. Efficiency and robustness of the overall method are assessed on the Burgers and Euler equations with shocks.

No abstract

This paper deals with intrusive Galerkin projection methods with Roe-type solver for uncertain hyperbolic systems using a finite volume discretization in physical space and a piecewise continuous representation at the stochastic level. The aim of this paper is to design a cost-effective adaptation of the deterministic Dubois and Mehlman corrector to avoid entropy-violating shocks in the presence of sonic points. The adaptation relies on an estimate of the eigenvalues and eigenvectors of the Galerkin Jacobian matrix of the deterministic system of the stochastic modes of the solution and on a correspondence between these approximate eigenvalues and eigenvectors for the intermediate states considered at the interface. Some indicators are derived to decide where a correction is needed, thereby reducing considerably the computational costs. The effectiveness of the proposed corrector is assessed on the Burgers and Euler equations including sonic points.

The multifield model like its counterpart, the two-fluid model, generally fails to be hyperbolic in its basic formulation. However, interfacial forces such as the interfacial pressure term and the virtual mass force, bringing new differential terms to the system, can change the analysis and make the problem hyperbolic. The aim of this paper is to define how the interfacial pressure default force affects the hyperbolicity of a general multifield model, and consequently, its inherent numerical stability. We characterize the location of the non hyperbolic regions and propose a closure for the interfacial pressure that makes the system hyperbolic in physically relevant regions.

No abstract

No abstract

No abstract

No abstract