*(English)*Zbl 1052.65114

Summary: An uncertainty quantification scheme is constructed based on generalized polynomial chaos representations. Two such representations are considered, based on the orthogonal projection of uncertain data and solution variables using either a Haar or a Legendre basis. Governing equations for the unknown coefficients in the resulting representations are derived using a Galerkin procedure and then integrated in order to determine the behavior of the stochastic process.

The schemes are applied to a model problem involving a simplified dynamical system and to the classical problem of Rayleigh-Bénard instability. For situations involving random parameters close to a critical point, the computational implementations show that the Wiener-Haar representation provides more robust predictions that those based on a Wiener-Legendre (WLe) decomposition. However, when the solution depends smoothly on the random data, the WLe scheme exhibits superior convergence. Suggestions regarding future extensions are finally drawn based on these experiences.

##### MSC:

65P20 | Numerical chaos |

65T60 | Wavelets (numerical methods) |

37C75 | Stability theory |

37D45 | Strange attractors, chaotic dynamics |

65P40 | Nonlinear stabilities (numerical analysis) |

37M25 | Computational methods for ergodic theory |