Date | 2015-11-13 |
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Speaker | Jae-Hun Jung |

Dept. | University at Buffalo, SUNY |

Room | 선택 |

Time | 15:00-16:00 |

* 장소: 25동 114호

Singular source terms play an important role in various PDEs such as the nonlinear Schr\"{o}dinger equation and the sine--Gordon equation. The singular source term in the equation mimics the local defect. The defect is highly localized and represented mathematically as a Dirac delta function that can be understood in the distribution sense. High-order approximations, such as the spectral approximation, of the solution with the singular source term may suffer from the Gibbs phenomenon. Moreover, if the singular source term is sensitive to the associated parameter values due to its locality and those parameters are stochastic, the high-order approximation needs a special treatment particularly to reduce the computational complexity. In this talk, we first explain the generalized polynomial chaos (gPC) method for solving time-dependent stochastic differential equations and related problems. For the Dirac delta function with the spectral method, we explain the consistent method based on the Schwartz duality of the delta distribution and use the consistent derivative(s) to the numerical scheme. We then, with the gPC method, solve the nonlinear Schr\"{o}dinger equation under the assumption that the singular source term is parameterized by random variables. For the solution in the random space, we adopt the orthogonal polynomials associated with the distribution that construct the random space. A numerical solution in the polynomial space is sought by the Galerkin procedure. Using the gPC method, we show that we can find the critical values efficiently and determine the statistical quantities with high accuracy

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