Nonlinear Model Order Reduction using Diffeomorphic Transformations of a Space-Time Domain

Kleikamp Hendrik, Ohlberger Mario, Rave Stephan

Research article in edited proceedings (conference) | Peer reviewed

Abstract

In many applications, for instance when describing dynamics of fluids or gases, hyperbolic conservation laws arise naturally in the modeling of conserved quantities of a system, like mass or energy. These types of equations exhibit highly nonlinear behaviors like shock formation or shock interaction. In the case of parametrized hyperbolic equations, where, for instance, varying transport velocities are considered, these nonlinearities and strong transport effects result in a highly nonlinear solution manifold. This solution manifold cannot be approximated properly by linear subspaces. To this end, nonlinear approaches for model order reduction of hyperbolic conservation laws are required. We propose a new method for nonlinear model order reduction that is especially well-suited for hyperbolic equations with discontinuous solutions. The approach is based on a space-time discretization and employs diffeomorphic transformations of the underlying space-time domain to align the discontinuities. To derive a reduced model for the diffeomorphisms, the Lie group structure of the diffeomorphism group is used to associate diffeomorphisms with corresponding velocity fields via the exponential map. In the linear space of velocity fields, standard model order reduction techniques, such as proper orthogonal decomposition, can be applied to extract a reduced subspace. For a parametrized Burgers' equation with two merging shocks, numerical experiments show the potential of the approach.

Details about the publication

PublisherBreitenecker, Felix; Kemmetmüller, Wolfgang; Körner, Andreas; Kugi, Andreas; Troch, Inge
Book titleMATHMOD 2022 - Discussion Contribution Volume (Volume 17)
Page range57-58
Publishing companyARGESIM Verlag
Place of publicationWien
Title of seriesARGESIM Report
Volume of series17
StatusPublished
Release year2022
Language in which the publication is writtenEnglish
ConferenceMATHMOD 2022, Wien, Austria
ISBN978-3-901608-95-7
DOI: 10.11128/arep.17.a17129
KeywordsModellreduktion; hyperbolische Erhaltungsgleichungen; Burgers Gleichung; Diffeomorphismen

Authors from the University of Münster

Kleikamp, Hendrik
Professorship of Applied Mathematics, especially Numerics (Prof. Ohlberger)
Ohlberger, Mario
Professorship of Applied Mathematics, especially Numerics (Prof. Ohlberger)
Center for Nonlinear Science
Center for Multiscale Theory and Computation
Rave, Stephan
Professorship of Applied Mathematics, especially Numerics (Prof. Ohlberger)