  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.
