  Intrusive uncertainty quantification methods for hyperbolic problems exhibit
spurious oscillations at shocks, which leads to a significant reduction of the
overall approximation quality. Furthermore, a challenging task is to preserve
hyperbolicity of the gPC moment system. An intrusive method which guarantees
hyperbolicity is the intrusive polynomial moment (IPM) method, which performs
the gPC expansion on the entropy variables. The method, while still being
subject to oscillations, requires solving a convex optimization problem in
every spatial cell and every time step. The aim of this work is to mitigate
oscillations in the IPM solution by applying filters. Filters reduce
oscillations by damping high order gPC coefficients. Naive filtering, however,
may lead to unrealizable moments, which means that the IPM optimization problem
does not have a solution and the method breaks down. In this paper, we propose
and analyze two separate strategies to guarantee the existence of a solution to
the IPM problem. First, we propose a filter which maintains realizability by
being constructed from an underlying Fokker-Planck equation. Second, we
regularize the IPM optimization problem to be able to cope with non-realizable
gPC coefficients. Consequently, standard filters can be applied to the
regularized IPM method. We demonstrate numerical results for the two strategies
by investigating the Euler equations with uncertain shock structures in one-
and two-dimensional spatial settings. We are able to show a significant
reduction of spurious oscillations by the proposed filters.

Numerical Analysis;Numerical Analysis