  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.
