Multi-Element Stochastic Galerkin Method Based on Edge Detection for Uncertainty Quantification of Discontinuous Responses

Abstract

We propose a new multi-element generalized polynomial chaos (MEgPC) method to minimize the computational costs required for the existing MEgPC to circumvent the Gibbs phenomenon in the presence of discontinuities in a random space. The proposed method uses edge detection to capture the discontinuous behavior of a solution with minimal decomposition. In contrast, the existing MEgPC iterates splitting the random space into two equal parts until achieving a sufficient resolution level. We take advantage of the fact that the stochastic Galerkin (SG) methods facilitate adaptive refinement of the decomposition at every time-step during a computation for the proposed method. The numerical experiments for two-test problems demonstrate the performance of the proposed method. The results show that the proposed method is consistently more accurate than conventional methods for sufficiently high polynomial orders with minimal additional computational costs to capture discontinuities.