超材料麦克斯韦方程组高精度能量守恒算法的理论及应用研究

For the supernormal electromagnetic features, metamaterials play very important roles in the fields of aerospace, military, national defense and communication. It is the most effective approach to study the electromagnetic properties of metamaterials by the behavior of the electromagnetic fields, which is computed through efficient simulations. Due to the micro-scales, refined and complicated structures and the long-time electromagnetic responses, simulations adapted to the large-scale and long-time computation on the refined meshes are confronted with great challenges. Therefore, developing efficient numerical methods which can also maintain the physical properties is urgent. Inspired by this, our project firstly obtains the energy-conserved properties for the electromagnetic fields in metamaterials and proposes the high-order energy-conserved numerical schemes for time domain Maxwell’s equations in high-dimensional metamaterials by operator splitting method and high-order finite difference method. Rigorous theoretical analysis is given. Secondly, for the geometries with curved surface, we extend the high-order conformal cell technique to the proposed high-order energy-conserved schemes. Furthermore, numerical algorithms with high accuracy for the scatterings of metamaterials are focused on by the methods of operator splitting, effective parameter and subgridding. Parallel software based on the proposed high-order algorithms is developed to compute the scattering of metamaterials. Further analysis of the scattering results helps to study the scattering characteristics of metamaterials. We are aiming to provide theoretical and technical support to the various applications of metamaterials.

超材料的反常电磁特性使其在航空航天、军事、国防、通信等领域具有非常广泛的应用前景。超材料电磁场计算作为超材料电磁特性研究的最有效途径，主要通过数值模拟来实现。微尺寸、精密复杂的几何结构以及长时间的电磁响应，使超材料电磁场的数值模拟面临在极细网格上进行长时间大规模计算的挑战，亟需开发高效快速且保持物理意义的数值方法。本项目从超材料麦克斯韦方程组的能量守恒性质出发，采用算子分裂、高阶差分等方法，构造高精度能量守恒的数值格式，进行理论分析；在此基础上，将高阶共形网格技术推广到所提出的高精度能量守恒格式中，构造曲面边界的高精度算法；研究超材料的电磁散射，结合算子分裂、等效参数、高阶子网格等方法和技术，提出高精度数值格式；开发超材料电磁散射的并行计算软件环境，分析其散射特性。本项目旨在为超材料在各应用领域的研究提供理论与技术支持。

超材料电磁场的高效数值模拟是超材料电磁特性理论及应用研究的关键基础。本课题重点研究高维超材料麦克斯韦方程组的高精度能量守恒算法、理论及应用，具有重要的科学意义和应用价值。一、我们在超材料电磁场计算的能量守恒算法方面取得重要进展，探究了Lorentz型超材料的电磁能量守恒性质，提出了求解三维Lorentz型超材料麦克斯韦方程组的时空二阶能量守恒分裂时域有限差分算法，证明了该算法的离散能量守恒、无条件稳定、时空二阶收敛性、超收敛性等，并将该算法应用于超材料散射问题的计算。二、我们在超材料电磁场计算的高精度能量守恒算法方面取得重要进展。一方面，为了提高空间精度，提出了求解二维Lorentz型超材料麦克斯韦方程组的紧致分裂时域有限差分算法，证明了算法的修正能量守恒、无条件稳定、时间二阶空间四阶收敛性，给出算法的数值色散关系式。另一方面，为了提高时间精度，我们将超材料麦克斯韦方程组的时空二阶交替方向隐式时域有限差分法作Richardson外推，得到具有时空四阶精度的算法，并从理论上严格证明了外推算法的时空收敛阶。课题研究成果可为超材料在各应用领域的研究提供理论与技术支持。