A Dynamic Bi-orthogonality based Approach for Uncertainty Quantification of Stochastic Systems with Discontinuities

The use of spectral projection based methods for simulation of a stochastic system with discontinuous solution exhibits the Gibbs phenomenon, which is characterized by oscillations near discontinuities. This paper investigates a dynamic bi-orthogonality based approach with appropriate post-processing for mitigating the effects of the Gibbs phenomenon. The proposed approach uses spectral decomposition of the spatial and stochastic fields in appropriate orthogonal bases, while the dynamic orthogonality condition is used to derive the resultant closed form evolution equations. The orthogonal decomposition of the spatial field is exploited to propose a Gegenbauer reprojection based post-processing approach, where the orthogonal bases in spatial dimension are reprojected on the Gegenbauer polynomials in the domain of analyticity. The resultant spectral expansion in Gegenbauer series is shown to mitigate the Gibbs phenomenon. Efficacy of the proposed method is demonstrated for simulation of a one-dimensional stochastic Burgers equation with uncertain initial condition.