A realizable filtered intrusive polynomial moment method

Abstract

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.

A Realizable Filtered Intrusive Polynomial Moment Method

“…In the following we will present numerical results for the two proposed filtering techniques. All results can be reproduced with the source code [32], which makes use of the intrusive UQ framework presented in [29].…”

“…Parts of this work have been discussed in the PhD thesis [31]. To allow reproducing the numerical results of this work, the source code of the IPM solver implementation is publicly available [32] together with scripts which recompute all test cases from this paper. The implementation is an extension of the code framework presented in [29].…”