Stochastic finite elements: a spectral approach
Stochastic finite elements: a spectral approach
A class of bases in L2 for the sparse representations of integral operators
SIAM Journal on Mathematical Analysis
The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
SIAM Journal on Scientific Computing
Uncertainty propagation using Wiener-Haar expansions
Journal of Computational Physics
Multi-resolution analysis of wiener-type uncertainty propagation schemes
Journal of Computational Physics
Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes
SIAM Journal on Scientific Computing
An adaptive multi-element generalized polynomial chaos method for stochastic differential equations
Journal of Computational Physics
Multi-Resolution-Analysis Scheme for Uncertainty Quantification in Chemical Systems
SIAM Journal on Scientific Computing
Uncertainty quantification for systems of conservation laws
Journal of Computational Physics
Padé-Legendre approximants for uncertainty analysis with discontinuous response surfaces
Journal of Computational Physics
Numerical analysis of the Burgers' equation in the presence of uncertainty
Journal of Computational Physics
Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems
Journal of Computational Physics
Roe solver with entropy corrector for uncertain hyperbolic systems
Journal of Computational and Applied Mathematics
Verification of variable-density flow solvers using manufactured solutions
Journal of Computational Physics
Hi-index | 31.45 |
The Euler equations subject to uncertainty in the initial and boundary conditions are investigated via the stochastic Galerkin approach. We present a new fully intrusive method based on a variable transformation of the continuous equations. Roe variables are employed to get quadratic dependence in the flux function and a well-defined Roe average matrix that can be determined without matrix inversion. In previous formulations based on generalized polynomial chaos expansion of the physical variables, the need to introduce stochastic expansions of inverse quantities, or square roots of stochastic quantities of interest, adds to the number of possible different ways to approximate the original stochastic problem. We present a method where the square roots occur in the choice of variables, resulting in an unambiguous problem formulation. The Roe formulation saves computational cost compared to the formulation based on expansion of conservative variables. Moreover, the Roe formulation is more robust and can handle cases of supersonic flow, for which the conservative variable formulation fails to produce a bounded solution. For certain stochastic basis functions, the proposed method can be made more effective and well-conditioned. This leads to increased robustness for both choices of variables. We use a multi-wavelet basis that can be chosen to include a large number of resolution levels to handle more extreme cases (e.g. strong discontinuities) in a robust way. For smooth cases, the order of the polynomial representation can be increased for increased accuracy.