Uniformly high order accurate essentially non-oscillatory schemes, 111
Journal of Computational Physics
Stochastic finite elements: a spectral approach
Stochastic finite elements: a spectral approach
On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation
Journal of Computational Physics
Journal of Computational Physics
Weighted essentially non-oscillatory schemes on triangular meshes
Journal of Computational Physics
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
An adaptive multi-element generalized polynomial chaos method for stochastic differential equations
Journal of Computational Physics
Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures
SIAM Journal on Scientific Computing
Essentially non-oscillatory Residual Distribution schemes for hyperbolic problems
Journal of Computational Physics
Wiener Chaos expansions and numerical solutions of randomly forced equations of fluid mechanics
Journal of Computational Physics
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
Journal of Computational Physics
A domain adaptive stochastic collocation approach for analysis of MEMS under uncertainties
Journal of Computational Physics
Time-dependent generalized polynomial chaos
Journal of Computational Physics
Spectral Methods for Parameterized Matrix Equations
SIAM Journal on Matrix Analysis and Applications
Subcell resolution in simplex stochastic collocation for spatial discontinuities
Journal of Computational Physics
A one-time truncate and encode multiresolution stochastic framework
Journal of Computational Physics
Hi-index | 31.46 |
This paper deals with the formulation of a semi-intrusive (SI) method allowing the computation of statistics of linear and non linear PDEs solutions. This method shows to be very efficient to deal with probability density function of whatsoever form, long-term integration and discontinuities in stochastic space. Given a stochastic PDE where randomness is defined on @W, starting from (i) a description of the solution in term of a space variables, (ii) a numerical scheme defined for any event @w@?@W and (iii) a (family) of random variables that may be correlated, the solution is numerically described by its conditional expectancies of point values or cell averages and its evaluation constructed from the deterministic scheme. One of the tools is a tessellation of the random space as in finite volume methods for the space variables. Then, using these conditional expectancies and the geometrical description of the tessellation, a piecewise polynomial approximation in the random variables is computed using a reconstruction method that is standard for high order finite volume space, except that the measure is no longer the standard Lebesgue measure but the probability measure. This reconstruction is then used to formulate a scheme on the numerical approximation of the solution from the deterministic scheme. This new approach is said semi-intrusive because it requires only a limited amount of modification in a deterministic solver to quantify uncertainty on the state when the solver includes uncertain variables. The effectiveness of this method is illustrated for a modified version of Kraichnan-Orszag three-mode problem where a discontinuous pdf is associated to the stochastic variable, and for a nozzle flow with shocks. The results have been analyzed in terms of accuracy and probability measure flexibility. Finally, the importance of the probabilistic reconstruction in the stochastic space is shown up on an example where the exact solution is computable, the viscous Burgers equation.