A discontinuous Galerkin method for viscous compressible multifluids

Abstract

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.

Quantum hydrodynamics with trajectories: The nonlinear conservation form mixed/discontinuous Galerkin method with applications in chemistry

“…Consider the following discretization scheme motivated by [13,30] (and illustrated in the one dimensional case in Fig. 3).…”

“…The discretization proceeds as in Sections 2 and 4, where we adopt the local Lax-Friedrich's advective flux with van Leer's MUSCL slope limiting scheme. Next we implement a standard explicit Runge-Kutta time discretization (see [10,35,30], or [29] for explicit details). Now we solve the resultant system using for our initial data (39), (40) explicitly that l ¼ 0:16; a ¼ 2; x 0 ¼ 3 and x 1 ¼ 6, such that, …”

Exactly solvable Madelung fluid and complex Burgers equations: a quantum Sturm–Liouville connection

“…In context of quantum hydrodynamics formulation it is possible to solve the chemical dynamics of several reaction mechanisms known to have pathways dominated by quantum tunneling regimes. These systems include proton transfer reactions, conformational inversions and proton-coupled electron transfer reactions [10]. The wave model approach to time dependent problems of quantum chemistry describes electrons as "clouds" moving in orbitals, and representing their positions by probability distribution.…”

Chemistry and Biochemistry

“…Then we can rewrite (2.1)-(2.2) as Consider the following discretization scheme motivated by [12,29] (and illustrated in the one dimensional case in Figure 3). Take an open Ω ⊂ R with boundary ∂Ω = Γ, given T > 0 such that Q T = ((0, T ) × Ω) forΩ the closure of Ω.…”

“…Next we implement a standard explicit Runge-Kutta time discretization (see [9,34] and [29], or [28] for explicit details). Now we solve the resultant system using for our initial data (5.…”