Stochastic approaches to uncertainty quantification in CFD simulations |
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Authors: | Lionel Mathelin M Yousuff Hussaini Thomas A Zang |
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Institution: | (1) Florida State University, 32306-4120 Tallahassee, FL, USA;(2) NASA Langley Research Center, 23681-2199 Hampton, VA, USA;(3) Present address: LIMSI-CNRS, Université Paris-Sud, Orsay, France |
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Abstract: | This paper discusses two stochastic approaches to computing the propagation of uncertainty in numerical simulations: polynomial
chaos and stochastic collocation. Chebyshev polynomials are used in both cases for the conventional, deterministic portion
of the discretization in physical space. For the stochastic parameters, polynomial chaos utilizes a Galerkin approximation
based upon expansions in Hermite polynomials, whereas stochastic collocation rests upon a novel transformation between the
stochastic space and an artificial space. In our present implementation of stochastic collocation, Legendre interpolating
polynomials are employed. These methods are discussed in the specific context of a quasi-one-dimensional nozzle flow with
uncertainty in inlet conditions and nozzle shape. It is shown that both stochastic approaches efficiently handle uncertainty
propagation. Furthermore, these approaches enable computation of statistical moments of arbitrary order in a much more effective
way than other usual techniques such as the Monte Carlo simulation or perturbation methods. The numerical results indicate
that the stochastic collocation method is substantially more efficient than the full Galerkin, polynomial chaos method. Moreover,
the stochastic collocation method extends readily to highly nonlinear equations. An important application is to the stochastic
Riemann problem, which is of particular interest for spectral discontinuous Galerkin methods. |
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Keywords: | uncertainty stochastic probabilistic polynomial chaos nozzle flow |
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