  We present a solution to the conservation form (Eulerian form) of the quantum
hydrodynamic equations which arise in chemical dynamics by implementing a
mixed/discontinuous Galerkin (MDG) finite element numerical scheme. We show
that this methodology is stable, showing good accuracy and a remarkable scale
invariance in its solution space. In addition the MDG method is robust,
adapting well to various initial-boundary value problems of particular
significance in a range of physical and chemical applications. We further show
explicitly how to recover the Lagrangian frame (or pathline) solutions.
