Supersensitivity due to uncertain boundary conditions. Xiu, D. & Karniadakis, G. E. International Journal for Numerical Methods in Engineering, 61(12):2114–2138, John Wiley & Sons, Ltd., 2004.
doi  abstract   bibtex   
We study the viscous Burgers' equation subject to perturbations on the boundary conditions. Two kinds of perturbations are considered: deterministic and random. For deterministic perturbations, we show that small perturbations can result in O(1) changes in the location of the transition layer. For random perturbations, we solve the stochastic Burgers' equation using different approaches. First, we employ the Jacobi-polynomial-chaos, which is a subset of the generalized polynomial chaos for stochastic modeling. Converged numerical results are reported (up to seven significant digits), and we observe similar ‘stochastic supersensitivity’ for the mean location of the transition layer. Subsequently, we employ up to fourth-order perturbation expansions. We show that even with small random inputs, the resolution of the perturbation method is relatively poor due to the larger stochastic responses in the output. Two types of distributions are considered: uniform distribution and a ‘truncated’ Gaussian distribution with no tails. Various solution statistics, including the spatial evolution of probability density function at steady state, are studied.
@Article{         Xiu_2004aa,
  abstract      = {We study the viscous Burgers' equation subject to perturbations on the boundary conditions. Two kinds of perturbations are considered: deterministic and random. For deterministic perturbations, we show that small perturbations can result in O(1) changes in the location of the transition layer. For random perturbations, we solve the stochastic Burgers' equation using different approaches. First, we employ the Jacobi-polynomial-chaos, which is a subset of the generalized polynomial chaos for stochastic modeling. Converged numerical results are reported (up to seven significant digits), and we observe similar ‘stochastic supersensitivity’ for the mean location of the transition layer. Subsequently, we employ up to fourth-order perturbation expansions. We show that even with small random inputs, the resolution of the perturbation method is relatively poor due to the larger stochastic responses in the output. Two types of distributions are considered: uniform distribution and a ‘truncated’ Gaussian distribution with no tails. Various solution statistics, including the spatial evolution of probability density function at steady state, are studied.},
  author        = {Xiu, Dongbin and Karniadakis, George Em},
  doi           = {10.1002/nme.1152},
  file          = {Xiu_2004aa.pdf},
  issn          = {1097-0207},
  journal       = {International Journal for Numerical Methods in Engineering},
  keywords      = {gpc,pde,sensitivity,random},
  langid        = {english},
  number        = {12},
  pages         = {2114--2138},
  publisher     = {John Wiley \& Sons, Ltd.},
  title         = {Supersensitivity due to uncertain boundary conditions},
  volume        = {61},
  year          = {2004},
  shortjournal  = {Int. J. Numer. Meth. Eng.}
}

Downloads: 0