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

Kleikamp Hendrik, Ohlberger Mario, Rave Stephan

Research article in edited proceedings (conference)

### 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 zur Publikation

Book title: MATHMOD 2022 - Discussion Contribution Volume

Release year: 2022

ISBN: 978-3-901608-95-7

Language in which the publication is written: English

Event: Wien