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.

翻译：本研究致力于发展熵稳定保正性间断Galerkin（DG）方法，作为模拟半导体能带中碰撞电子输运概率密度的Boltzmann-Poisson系统的计算方案。在表示球面/能量-角度变量的动量坐标中，我们建立了相应的Vlasov-Boltzmann方程，该方程采用线性碰撞算子和奇异测度，其散射模型作为能带结构的函数适用于热电子纳米尺度输运。利用碰撞算子熵不等式所赋予的耗散特性，我们证明了一维位置（二维动量）和二维位置（三维动量）情况下半离散DG格式在熵范数下的稳定性。该熵依赖于哈密顿量的指数函数，而非仅与动能相关的麦克斯韦分布。对于一维问题，掌握泊松方程的解析解以及电流收敛至常数的特性，对于获得完全稳定性（加权熵范数随时间递减）至关重要。对于二维问题，在估计熵范数下的稳定性时考虑了镜面反射边界条件和周期性条件。关于一维问题DG格式的保正性证明，受\cite{ZhangShu1}、\cite{ZhangShu2}、\cite{CGP}和\cite{EECHXM-JCP}的启发，我们将碰撞视为源项，并找到输运项与碰撞项的凸组合，从而保证下一时刻数值概率密度单元平均值的正性。通过应用\cite{ZhangShu1}和\cite{ZhangShu2}中保持单元平均值的限制器，修正分段线性解的斜率以确保函数非负，从而保证计算域内概率密度数值解的正性。