Oscillation Mitigation of Hyperbolicity-Preserving Intrusive Uncertainty Quantification Methods for Systems of Conservation Laws

TL;DR

This paper develops and compares methods to reduce oscillations and preserve hyperbolicity in intrusive uncertainty quantification for systems of conservation laws, improving stability and computational efficiency.

Contribution

It introduces filtering, hyperbolicity limiters, and multi-element polynomial moment methods to mitigate oscillations and reduce costs in hyperbolic PDE uncertainty quantification.

Findings

Filtering reduces oscillations in stochastic Galerkin schemes.

Multi-element IPM decreases computational costs and enhances parallelization.

Numerical examples show mitigation of artifacts and improved efficiency.

Abstract

In this article we study intrusive uncertainty quantification schemes for systems of conservation laws with uncertainty. Standard intrusive methods lead to oscillatory solutions which sometimes even cause the loss of hyperbolicity. We consider the stochastic Galerkin scheme, in which we filter the coefficients of the polynomial expansion in order to reduce oscillations. We further apply the multi-element approach and ensure the preservation of hyperbolic solutions through the hyperbolicity limiter. In addition to that, we study the intrusive polynomial moment method, which guarantees hyperbolicity at the cost of solving an optimization problem in every spatial cell and every time step. To reduce numerical costs, we apply the multi-element ansatz to IPM. This ansatz decouples the optimization problems of all multi elements. Thus, we are able to significantly decrease computational costs while improving parallelizability. We finally evaluate these oscillation mitigating approaches on various numerical examples such as a NACA airfoil and a nozzle test case for the two-dimensional Euler equations. In our numerical experiments, we observe the mitigation of spurious artifacts. Furthermore, using the multi-element ansatz for IPM significantly reduces computational costs.