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： 本工作开发了熵稳定且保持正性的DG方法，作为计算方案用于模拟半导体晶体晶格中沿能带的电子输运的概率密度函数（pdf）的玻尔兹曼-泊松系统。 我们使用球坐标或能量角变量作为动量坐标，提出了相应的Vlasov玻尔兹曼方程。 该方程具有一个线性碰撞算子，其奇异测度模型将散射表示为能带的函数。 我们展示了半离散DG格式在熵范数下的稳定性结果，适用于1D位置2D动量和2D位置3D动量的情况，利用了碰撞算子的耗散性质，其熵不等式依赖于整个哈密顿量，而不仅仅是动能。 对于1D问题，了解泊松方程的解析解以及收敛到恒定电流的知识对于获得完全稳定性至关重要。 对于2D问题，在熵范数下稳定性估计中除了周期性外还考虑了镜面反射边界条件。 关于保持正性（1D位置），我们将碰撞算子视为源项，并找到运输项和碰撞项的凸组合，以保证下一时间步数值pdf的单元平均值的正性。 通过应用自然限制器，可以保证整个域中数值pdf的正性，这些限制器保持单元平均值不变，但修改分段线性解的斜率，以使函数非负。 使用球坐标系$\vec{p}(|\vec{p}|,\mu=cos\theta,\varphi)$与之前BP的DG求解器中的选择略有不同，因为所提出的DG公式在运输和碰撞项中涉及仅包含分段多项式函数的积分，这比之前的方法更适合高斯求积。 显示英文摘要

摘要： 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.