A dynamically adaptive wavelet approach to stochastic computations based on polynomial chaos - capturing all scales of random modes on independent grids

  • Authors:

  • Xiaoan Ren;Wenquan Wu;Leonidas S. Xanthis

  • Affiliations:

  • University of Shanghai for Science and Technology, Shanghai 200093, PR China;University of Shanghai for Science and Technology, Shanghai 200093, PR China;Department of Aeronautics, Imperial College London, Prince Concort Road, South Kensington Campus, London SW7 2BY, UK

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2011

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Abstract

In stochastic computations, or uncertainty quantification methods, the spectral approach based on the polynomial chaos expansion in random space leads to a coupled system of deterministic equations for the coefficients of the expansion. The size of this system increases drastically when the number of independent random variables and/or order of polynomial chaos expansions increases. This is invariably the case for large scale simulations and/or problems involving steep gradients and other multiscale features; such features are variously reflected on each solution component or random/uncertainty mode requiring the development of adaptive methods for their accurate resolution. In this paper we propose a new approach for treating such problems based on a dynamically adaptive wavelet methodology involving space-refinement on physical space that allows all scales of each solution component to be refined independently of the rest. We exemplify this using the convection-diffusion model with random input data and present three numerical examples demonstrating the salient features of the proposed method. Thus we establish a new, elegant and flexible approach for stochastic problems with steep gradients and multiscale features based on polynomial chaos expansions.