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MSC: 35Q30 35Q80 76D05 76D06 35K57 65M25 65M60 65N30 65N25 76T30 76T10 76N15 Keywords: Navier-Stokes Discontinuous Galerkin Numerical methods Runge-Kutta Mixing Barotropic Compressible flow Viscous Miscible Multicomponent Multiphase Multifluid Chemical Acoustic a b s t r a c tWe present a generalized discontinuous Galerkin method for a multicomponent compressible barotropic Navier-Stokes system of equations. The system presented has a functional viscosity m which depends on the pressure p ¼ pðq; l i Þ of the flow, with the density q and the local concentration l i . High order Runge-Kutta time-discretization techniques are employed, and different methods of dealing with arbitrary coupled boundary conditions are discussed. Analysis of the energy consistency of the scheme is performed in addition to inspection of the relative error of the solution compared to exact analytic test cases.Finally several examples, comparisons, generalizations and physical applications are presented.

a b s t r a c tWe 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.

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