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A WENO-solver combined with adaptive momentum discretization for the Wigner transport equation and its application to resonant tunneling diodes

机译：WENO求解器与自适应动量离散化相结合的Wigner输运方程及其在共振隧穿二极管中的应用

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摘要

We present a novel numerical scheme for the deterministic solution of the Wigner transport equation, especially suited to deal with situations in which strong quantum effects are present. The unique feature of the algorithm is the expansion of the Wigner function in local basis functions, similar to finite element or finite volume methods. This procedure yields a discretization of the pseudo-differential operator that conserves the particle density on arbitrarily chosen grids. The high flexibility in refining the grid spacing together with the weighted essentially non-oscillatory (WENO) scheme for the advection term allows for an accurate and well-resolved simulation of the phase space dynamics. A resonant tunneling diode is considered as test case and a detailed convergence study is given by comparing the results to a non-equilibrium Green's functions calculation. The impact of the considered domain size and of the grid spacing is analyzed. The obtained convergence of the results towards a quasi-exact agreement of the steady state Wigner and Green's functions computations demonstrates the accuracy of the scheme, as well as the high flexibility to adjust to different physical situations.
机译：我们为维格纳输运方程的确定性解决方案提供了一种新颖的数值方案，特别适合处理存在强量子效应的情况。该算法的独特之处在于Wigner函数在局部基函数中的扩展，类似于有限元或有限体积方法。该过程产生伪微分算子的离散化，该伪微分算子在任意选择的网格上保留了粒子密度。细化网格间距的高度灵活性以及对流项的加权基本非振荡（WENO）方案，可以对相空间动力学进行精确且良好解析的模拟。谐振隧道二极管被认为是测试案例，通过将结果与非平衡格林函数计算进行比较，给出了详细的收敛性研究。分析了考虑的域大小和网格间距的影响。所获得的结果趋近于稳态Wigner和Green函数计算的准精确一致，证明了该方案的准确性以及适应不同物理情况的高度灵活性。

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