A monotone numerical flux for quasilinear convection diffusion equation

We propose a new numerical 2-point flux for quasilinear convection–diffusion equation. This is shown to be an approximation of the derived from solution two-point Dirichlet boundary value problem projection continuous onto line connecting neighboring collocation points. The later approach generalizes idea first proposed by Scharfetter and Gummel [IEEE Trans. Electron Devices 16 (1969), pp. 64–77] linear drift-diffusion equations. establish that satisfies sufficient properties ensuring convergence associate finite volume scheme, while respecting maximum principle. Then, we pay attention long time behavior scheme: show relative entropy decay satisfied as well generalized Scharfetter-Gummel flux. proof these uses generalization some discrete (and continuous) log-Sobolev inequalities. corresponding proved in appendix. Some 1D experiments confirm theoretical results.