ISBN 978-3-8439-4731-2

978-3-8439-4731-2, Reihe Mathematik

Louisa Schlachter
Stochastic Galerkin Methods in Hyperbolic Equations

153 Seiten, Dissertation Technische Universität Kaiserslautern (2021), Softcover, A5

Zusammenfassung / Abstract

Uncertainty quantification through stochastic spectral methods is rising in popularity. In this thesis we extensively study the intrusive stochastic Galerkin scheme for uncertain systems of conservation laws.

The classical SG method is well-known to produce Gibbs oscillations arising from interpolating or projecting discontinuous data. Thus, the approach is not ensured to preserve hyperbolicity of the underlying hyperbolic system of equations. Therefore, we develop a hyperbolicity-preserving modification of SG, using a slope limiter to retain admissible solutions. In order to accomplish high-order approximations, we combine the method with a Runge-Kutta discontinuous Galerkin scheme and a multi-element SG ansatz in the random space. We perform convergence studies for the modified numerical scheme and apply it to different challenging numerical examples for which the classical SG approach fails.

We deduce further modifications of SG since the method still appears to produce oscillations already for a simple example of uncertain linear advection. We introduce the so-called WENO SG scheme, which is constructed to prevent the propagation of Gibbs phenomenon by applying a slope limiter in the stochasticity and WENO reconstructions in the spatial and stochastic domain.

Furthermore, we include SG into a comparison of intrusive and non-intrusive UQ schemes. In particular, the performance of SG in multiple stochastic dimensions is compared to stochastic collocation in order to find the range of dimensions in which SG is superior to non-intrusive methods. We analyze the differences from SG to multi-level Monte Carlo and the intrusive polynomial moment method, resulting in a discussion about the applicability of our UQ approaches.