  The solution of conservation laws with parametrized shock waves presents
challenges for both high-order numerical methods and model reduction
techniques. We introduce an r-adaptivity scheme based on optimal transport and
apply it to develop reduced order models for compressible flows. The optimal
transport theory allows us to compute high-order r-adaptive meshes from a
starting reference mesh by solving the Monge-Ampere equation. A high-order
discretization of the conservation laws enables high-order solutions to be
computed on the resulting r-adaptive meshes. Furthermore, the Monge-Ampere
solutions contain mappings that are used to reduce the spatial locality of the
resulting solutions and make them more amenable to model reduction. We use a
non-intrusive model reduction method to construct reduced order models of both
the mesh and the solution. The procedure is demonstrated on three supersonic
and hypersonic test cases, with the hybridizable discontinuous Galerkin method
being used as the full order model.
