标题： 熵稳定保持正性的DG方案用于沿能带的碰撞电子输运的玻尔兹曼-泊松模型 显示英文标题

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

摘要： 这项工作涉及开发熵稳定且保持正性的不连续伽辽金（DG）方法，作为用于模拟半导体能带中碰撞电子输运概率密度的玻尔兹曼-泊松系统的计算方案。在表示球形/能量角变量的动量坐标中，我们提出了相应的Vlasov-Boltzmann方程，其中包含一个线性碰撞算子和一个奇异测度，以适当的方式将散射建模为带结构的函数，适用于热电子纳米尺度输运。我们展示了在熵范数下半离散DG格式的稳定性结果，针对一维位置（二维动量）和二维位置（三维动量），利用碰撞算子的耗散性质及其熵不等式。后者依赖于哈密顿量的指数，而不是仅与动能相关的麦克斯韦分布。对于一维问题，知道泊松方程的解析解并收敛到恒定电流对于获得完全稳定性（加权熵范数随时间减少）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。关于DG方案中一维问题的保持正性证明，受\cite{ZhangShu1}、\cite{ZhangShu2}和\cite{CGP}、\cite{EECHXM-JCP}的启发，我们将碰撞视为源项，并找到传输项和碰撞项的凸组合，以保证下一时间步数值概率密度的单元平均值的正性。通过应用保留单元平均值的限制器\cite{ZhangShu1}和\cite{ZhangShu2}，修改分段线性解的斜率，从而保证域内概率密度数值解的正性。 显示英文摘要

摘要： 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.