Instabilities of electron plasma waves in high-mobility semiconductor devices have recently attracted a lot of attention as a possible candidate for closing the THz gap. Conventional moments-based transport models usually neglect time derivatives in the constitutive equations for vectorial quantities, resulting in parabolic systems of partial differential equations (PDE). To describe plasma waves however, such time derivatives need to be included, resulting in hyperbolic rather than parabolic systems of PDEs; thus the fundamental nature of these equation systems is changed completely. Additional nonlinear terms render the existing numerical stabilization methods for semiconductor simulation practically useless. On the other hand there are plenty of numerical methods for hyperbolic systems of PDEs in the form of conservation laws. Standard numerical schemes for conservation laws, however, are often either incapable of correctly handling the large source terms present in semiconductor devices due to built-in electric fields, or rely heavily on variable transformations which are specific to the equation system at hand (e.g. the shallow water equations), and can not be generalized easily to different equations. In this paper we develop a novel well-balanced numerical scheme for hyperbolic systems of PDEs with source terms and apply it to a simple yet non-linear electron transport model.

高迁移率半导体器件中电子等离子体波的不稳定性最近作为引起太赫兹（THz）差距缩小的候选者而引起了广泛关注。传统的基于矩的传输模型通常忽略矢量量的本构方程中的时间导数，从而形成了偏微分方程（PDE）的抛物线系统。然而，为了描述等离子波，需要包括这样的时间导数，从而导致PDE的双曲系统而不是抛物线系统。因此，这些方程组的基本性质被完全改变。附加的非线性项使得现有的用于半导体仿真的数值稳定方法几乎无用。另一方面，对于双曲偏微分方程的双曲系统，存在许多守恒律形式的数值方法。但是，用于守恒定律的标准数值方案通常要么不能正确处理由于内置电场导致的半导体器件中存在的大量源项，要么在很大程度上依赖于特定方程组的变量转换（例如，浅水方程），并且不能轻易地推广到不同的方程。在本文中，我们为带有源项的PDE双曲系统开发了一种新型的平衡良好的数值方案，并将其应用于简单但非线性的电子传输模型。浅水方程），并且不能轻易地推广到不同的方程。在本文中，我们为带有源项的PDE双曲系统开发了一种新型的平衡良好的数值方案，并将其应用于简单但非线性的电子传输模型。浅水方程），并且不能轻易地推广到不同的方程。在本文中，我们为带有源项的PDE双曲系统开发了一种新型的平衡良好的数值方案，并将其应用于简单但非线性的电子传输模型。