Entropy-stable positivity-preserving DG schemes for Boltzmann-Poisson models of collisional electronic transport along energy bands
This work is related to the development of entropy-stable positivity-preserving DG methods as a computational scheme for Boltzmann-Poisson systems modeling the probability density of collisional electronic transport along energy bands in semiconductors. We pose, in momentum coordinates representing spherical / energy-angular variables, the respective Vlasov-Boltzmann eq. with a linear collision operator and a singular measure, modeling scatterings as functions of the bandstructure appropriately for hot electron nanoscale transport. We show stability results of semidiscrete DG schemes under an entropy norm for 1D, 2D position, 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. For the 2D problem, specular reflection boundary conditions and periodicity are considered in estimating stability under an entropy norm. Regarding the positivity-preservation in the DG scheme for the 1D problem, 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 well-known limiters that preserve the cell average modifying the slope of the piecewise linear solutions to make the function non-negative. The use of a spherical coordinate system whose radial component is the momentum magnitude is slightly different from choices in previous DG solvers for Boltzmann-Poisson, since the proposed DG formulation gives simpler integrals involving piecewise polynomials.