基于位函数的统一辛时域有限差分算法及其在多物理仿真中的应用

Advances in nanofabrication techniques have made it possible to achieve ultrasmall nanodevices with intricate structure. In this situation, the problem of low-frequency collapse, low computation accuracy and computational efficiency and the intrinsic quantum effects should be considered in the traditional time-domain electromagnetic computational method. The proposed subject aims at high accuracy and efficiency solution for the Maxwell-Schrödinger multiphysics equation based on A-Phi potential function. The related research content includes: (1) Starting from the potential function and using the Lorenz specification, constructing a unified symplectic finite-difference time-domain algorithm for the coupled systems. (2) Studying the key techniques of unified finite-difference time-domain algorithm, such as subgrid technology, unconditionally stable method and absorption boundary condition and so on. (3) In view of the specific multi-physics problems, according to the physical meaning of the method itself, the system coupling modeling method under other specifications is studied. Research on the coupling modeling method of the system under other guage conditions, seek a unified approach for multi-field applications based on potential functions will be well investigated. The theory and its algorithm are applied to the simulation of complex structure, and can be used as a theoretical basis and technical support for the design of the new type of micro/nano structures and the development of the computational electromagnetics.

针对传统电磁场时域方法在计算大规模纳米结构或包含极精细结构时面临的低频崩溃、计算精度低和计算效率差的难题以及存在的量子效应问题，项目拟研究基于位函数A-Phi的高精度、高效率的麦克斯韦-薛定谔方程耦合算法。具体研究内容包括（1）从位函数出发，利用洛伦兹规范，构建耦合系统统一的辛时域有限差分算法；（2）构建统一辛时域有限差分算法中的关键技术如亚网格技术、无条件稳定方法及吸收边界条件等；（3）针对具体的多物理问题，根据方法自身的物理意义，研究在其他规范条件下系统的耦合建模方法，寻求基于位函数的适合多领域应用的统一方法。将理论及其算法应用于复杂媒质结构模拟中，为新型微纳结构的设计及计算电磁学的发展提供理论依据和技术支持。

针对传统电磁场时域方法在计算大规模纳米结构或包含极精细结构时面临的低频崩溃、计算精度低和计算效率差及存在的量子效应难题，项目研究了一种基于位函数A-Phi的高精度、高效率的麦克斯韦-薛定谔方程耦合算法。针对上述问题，具体研究内容和成果如下：.（1）从位函数出发，利用洛伦兹规范，构建了耦合系统的统一辛时域有限差分算法，用于模拟电磁场-粒子相互作用的麦克斯韦-薛定谔耦合系统，并建立了高阶辛时域有限差分算法的离散框架；.（2）开展了辛时域有限差分算法关键技术研究，构建了导波结构的高阶辛紧有限时域差分算法。相比较于传统紧时域有限差分(FDTD)算法，该算法可以显著减小数值色散误差，改善了Courant-Friedrich-Levy限制条件；.（3）针对多物理问题，研究在其他规范条件下系统的耦合建模方法。针对非均匀电磁系统，研究了通用矢量-标量势框架及其在半经典量子电磁学中的应用。以不同规范条件下E-B-A -φ公式为基础，提出了一般电磁系统的电磁参数的统一化数值解，并采用FDTD方法对这些公式进行离散化，为新型微纳结构的设计及计算电磁学的发展提供理论依据和技术支持。