Entropy-stable positivity-preserving DG schemes for Boltzmann-Poisson models of collisional electronic transport along energy bands

This work is related to developing entropy-stable positivity-preserving Discontinuous Galerkin (DG) methods as a computational scheme for Boltzmann-Poisson systems modeling the probability density of collisional electronic transport along semiconductor energy bands. In momentum coordinates representing spherical / energy-angular variables, we pose the respective Vlasov-Boltzmann equation with a linear collision operator and a singular measure, modeling scatterings as functions of the band structure appropriately for hot electron nanoscale transport. We show stability results of semi-discrete DG schemes under an entropy norm for 1D-position (2D-momentum) and 2D-position (3D-momentum), using dissipative properties of the collisional operator given its entropy inequality. The latter depends on an exponential of the Hamiltonian rather than the Maxwellian associated with only kinetic energy. For the 1D problem, knowing the analytic solution to the Poisson equation and convergence to a constant current is crucial to obtaining full stability (weighted entropy norm decreasing over time). For the 2D problem, specular reflection boundary conditions and periodicity are considered in estimating stability under an entropy norm. Regarding the positivity-preservation proofs in the DG scheme for the 1D problem, inspired by \cite{ZhangShu1}, \cite{ZhangShu2}, and \cite{CGP}, \cite{EECHXM-JCP}, we treat collisions as a source and find convex combinations of the transport and collision terms which guarantee positivity of the cell average of our numerical probability density at the next time. The positivity of the numerical solution to the probability density in the domain is guaranteed by applying the limiters in \cite{ZhangShu1} and \cite{ZhangShu2} that preserve the cell average modifying the slope of the piecewise linear solutions to make the function non-negative.