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A Spectral Stochastic Semi-Lagrangian Method for Convection-Diffusion Equations with Uncertainty

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Abstract

Real life convection-diffusion problems are characterized by their inherent or externally induced uncertainties in the design parameters. This paper presents a spectral stochastic finite element semi-Lagrangian method for numerical solution of convection-diffusion equations with uncertainty. Using the spectral decomposition, the stochastic variational problem is reformulated to a set of deterministic variational problems to be solved for each Wiener polynomial chaos. To obtain the chaos coefficients in the corresponding deterministic convection-diffusion equations, we implement a semi-Lagrangian method in the finite element framework. Once this representation is computed, statistics of the numerical solution can be easily evaluated. These numerical techniques associate the geometrical flexibility of the finite element method with the ability offered by the semi-Lagrangian method to solve convection-dominated problems using time steps larger than its Eulerian counterpart. Numerical results are shown for a convection-diffusion problem driven with stochastic velocity and for an incompressible viscous flow problem with a random force. In both examples, the proposed method demonstrates its ability to better maintain the shape of the solution in the presence of uncertainties and steep gradients.

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Authors and Affiliations

  1. Dpto. Matemática Aplicada, Universidad Rey Juan Carlos, 28933, Móstoles-Madrid, Spain

    Mofdi El-Amrani

  2. School of Engineering, University of Durham, South Road, Durham, DH1 3LE, UK

    Mohammed Seaïd

Authors
  1. Mofdi El-Amrani
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  2. Mohammed Seaïd
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Correspondence to Mofdi El-Amrani.

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El-Amrani, M., Seaïd, M. A Spectral Stochastic Semi-Lagrangian Method for Convection-Diffusion Equations with Uncertainty. J Sci Comput 39, 371–393 (2009). https://doi.org/10.1007/s10915-009-9273-5

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  • DOI: https://doi.org/10.1007/s10915-009-9273-5

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