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We propose a new numerical 2-point flux for a quasilinear convection-diffusion equation. This numerical flux is shown to be an approximation of the numerical flux derived from the solution of a two-point Dirichlet boundary value problem for the projection of the continuous flux onto the line connecting neighboring collocation points. The later approach generalizes an idea first proposed by Scharfetter and Gummel for linear drift-diffusion equations. We establish first that the new flux satisfies sufficient properties ensuring the convergence of the associate finite volume scheme, while respecting the maximum principle. Then, we pay attention to the long time behavior of the scheme: we show relative entropy decay properties satisfied by the new numerical flux as well as by the generalized Scharfetter-Gummel flux. The proof of these properties uses a generalization of some discrete (and continuous) log-Sobolev inequalities. The corresponding decay of the relative entropy of the continuous solution is proved in the appendix. Some 1D numerical experiments confirm the theoretical results.

Despite the fact that the theory of mixtures has been part of non-equilibrium thermodynamics and engineering for a long time, it is far from complete. While it is well formulated and tested in the case of mechanical equilibrium (where only diffusion-like processes take place), the question how to properly describe homogeneous mixtures that flow with multiple independent velocities that still possess some inertia (before mechanical equilibrium is reached) is still open. Moreover, the mixtures can have several temperatures before they relax to a common value. In this paper, we derive a theory of mixtures from Hamiltonian mechanics in interaction with electromagnetic fields. The resulting evolution equations are then reduced to the case with only one momentum (classical irreversible thermodynamics), providing a generalization of the Maxwell-Stefan diffusion equations. In a next step, we reduce that description to the mechanical equilibrium (no momentum) and derive a non-isothermal variant of the dusty gas model. These reduced equations are solved numerically, and we illustrate the results on efficiency analysis, showing where in a concentration cell efficiency is lost. Finally, the theory of mixtures identifies the temperature difference between constituents as a possible new source of the Soret coefficient. For the sake of clarity, we restrict the presentation to the case of binary mixtures; the generalization is straightforward.

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In this paper, we consider a drift-diffusion system with cross-coupling through the chemical potentials comprising a model for the motion of finite size ions in liquid electrolytes. The drift term is due to the self-consistent electric field maintained by the ions and described by a Poisson equation. We design two finite volume schemes based on different formulations of the fluxes. We also provide a stability analysis of these schemes and an existence result for the corresponding discrete solutions. A convergence proof is proposed for non-degenerate solutions. Numerical experiments show the behavior of these schemes.

We compare three thermodynamically consistent numerical fluxes known in the literature, appearing in a Voronoi finite volume discretization of the van Roosbroeck system with general charge carrier statistics. Our discussion includes an extension of the Scharfetter–Gummel scheme to non-Boltzmann (e.g. Fermi–Dirac) statistics. It is based on the analytical solution of a two-point boundary value problem obtained by projecting the continuous differential equation onto the interval between neighboring collocation points. Hence, it serves as a reference flux. The exact solution of the boundary value problem can be approximated by computationally cheaper fluxes which modify certain physical quantities. One alternative scheme averages the nonlinear diffusion (caused by the non-Boltzmann nature of the problem), another one modifies the effective density of states. To study the differences between these three schemes, we analyze the Taylor expansions, derive an error estimate, visualize the flux error and show how the schemes perform for a carefully designed p-i-n benchmark simulation. We present strong evidence that the flux discretization based on averaging the nonlinear diffusion has an edge over the scheme based on modifying the effective density of states.