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

This work develops entropy-stable positivity-preserving DG methods as a computational scheme for Boltzmann-Poisson systems modeling the pdf of electronic transport along energy bands in semiconductor crystal lattices. We pose, using spherical or energy-angular variables as momentum coordinates, the corresponding Vlasov Boltzmann eq. with a linear collision operator with a singular measure modeling the scattering as functions of the energy band. We show stability results of semi-discrete DG schemes under an entropy norm for 1D-position 2D-momentum, and 2D-position 3D-momentum, using the dissipative properties of the collisional operator given its entropy inequality, which depends on the whole Hamiltonian rather than only the kinetic energy. For the 1D problem, knowledge of the analytic solution to Poisson and of the convergence to a constant current is crucial to obtain full stability. For the 2D problem, specular reflection BC are considered in addition to periodicity in the estimate for stability under an entropy norm. Regarding positivity preservation (1D position), we treat the collision operator as a source term and find convex combinations of the transport and collision terms which guarantee the positivity of the cell average of our numerical pdf at the next time step. The positivity of the numerical pdf in the whole domain is guaranteed by applying the natural limiters that preserve the cell average but modify the slope of the piecewise linear solutions in order to make the function non-negative. The use of a spherical coordinate system $\vec{p}(|\vec{p}|,\mu=cos\theta,\varphi)$ is slightly different to the choice in previous DG solvers for BP, since the proposed DG formulation gives simpler integrals involving just piecewise polynomial functions for both transport and collision terms, which is more adequate for Gaussian quadrature than previous approaches.