  Proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD)
are two complementary singular-value decomposition (SVD) techniques that are
widely used to construct reduced-order models (ROMs) in a variety of fields of
science and engineering. Despite their popularity, both DMD and POD struggle to
formulate accurate ROMs for advection-dominated problems because of the nature
of SVD-based methods. We investigate this shortcoming of conventional POD and
DMD methods formulated within the Eulerian framework. Then we propose a
Lagrangian-based DMD method to overcome this so-called translational issues.
Our approach is consistent with the spirit of physics-aware DMD since it
accounts for the evolution of characteristic lines. Several numerical tests are
presented to demonstrate the accuracy and efficiency of the proposed Lagrangian
DMD method.
