Stochastic finite elements: a spectral approach
Stochastic finite elements: a spectral approach
A stochastic projection method for fluid flow. I: basic formulation
Journal of Computational Physics
The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
SIAM Journal on Scientific Computing
A stochastic projection method for fluid flow II.: random process
Journal of Computational Physics
Uncertainty propagation using Wiener-Haar expansions
Journal of Computational Physics
Multi-resolution analysis of wiener-type uncertainty propagation schemes
Journal of Computational Physics
Natural Convection in a Closed Cavity under Stochastic Non-Boussinesq Conditions
SIAM Journal on Scientific Computing
Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes
SIAM Journal on Scientific Computing
High-Order Collocation Methods for Differential Equations with Random Inputs
SIAM Journal on Scientific Computing
Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures
SIAM Journal on Scientific Computing
Predicting shock dynamics in the presence of uncertainties
Journal of Computational Physics - Special issue: Uncertainty quantification in simulation science
Multi-Resolution-Analysis Scheme for Uncertainty Quantification in Chemical Systems
SIAM Journal on Scientific Computing
Sparse grid collocation schemes for stochastic natural convection problems
Journal of Computational Physics
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
The multi-element probabilistic collocation method (ME-PCM): Error analysis and applications
Journal of Computational Physics
Uncertainty quantification for systems of conservation laws
Journal of Computational Physics
Roe solver with entropy corrector for uncertain hyperbolic systems
Journal of Computational and Applied Mathematics
Roe solver with entropy corrector for uncertain hyperbolic systems
Journal of Computational and Applied Mathematics
Generalised Polynomial Chaos for a Class of Linear Conservation Laws
Journal of Scientific Computing
Uncertainty quantification in kinematic-wave models
Journal of Computational Physics
Uncertainty quantification in hybrid dynamical systems
Journal of Computational Physics
Simplex stochastic collocation with ENO-type stencil selection for robust uncertainty quantification
Journal of Computational Physics
Subcell resolution in simplex stochastic collocation for spatial discontinuities
Journal of Computational Physics
A stochastic Galerkin method for the Euler equations with Roe variable transformation
Journal of Computational Physics
High-order methods as an alternative to using sparse tensor products for stochastic Galerkin FEM
Computers & Mathematics with Applications
Segmentation of Stochastic Images using Level Set Propagation with Uncertain Speed
Journal of Mathematical Imaging and Vision
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This 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.