In this work, we investigate the propagation of electromagnetic waves in the Cole-Cole dispersive medium by using the discontinuous Galerkin (DG) method to solve the coupled time-domain Maxwell's equations and polarization equation. We define a new and sharpened total energy function for the Cole-Cole model, which better describes the behaviors of the energy than what is available in the current literature. A major theme in the time-domain numerical modeling of this problem has been tackling the difficulty of handling the nonlocal term involved in the time-domain polarization equation. Based on the diffusive representation and the quadrature formula, we derive an approximate system, where the convolution kernel is replaced by a finite number of auxiliary variables that satisfy local-in-time ordinary differential equations. To ensure the resulted approximate system is stable, a nonlinear constrained optimization numerical scheme is established to determine the quadrature coefficients. By a special choice of the numerical fluxes and projections, we obtain {for the constant coefficient case } an optimal-order convergence result for the semi-discrete DG scheme. The temporal discretization is achieved by the standard two-step backward difference formula and a fast algorithm with linear complexity is constructed. Numerical examples are provided for demonstrating the efficiency of the proposed algorithm, validating the theoretical results and illustrating the behaviors of the energy.
翻译:在这项工作中,我们通过使用不连续的 Galerkin (DG) 方法解决时间- Cole 分散介质中电磁波的传播问题。 我们为Cole- Cole 模型定义了一个新的和更加精锐的总能量功能,该模型比当前文献中可以提供的更好地描述能源行为。 时间- 数字模型中的一个主要主题一直是解决处理时间- 度极化等式所涉及的非本地术语的困难。 根据调控表达式和等离子公式,我们产生了一个近似系统,在这个系统中,共振内核被数量有限的辅助变量所取代,这些变量能满足当地- 时间普通差异方程式。为了确保结果的近似系统稳定,将建立一个非线性限制优化的数值方案,以确定四分率系数。 通过对数值通量和预测的特殊选择,我们获得了 用于半偏移的系数和等式公式的优化组合结果。 我们获得了一个半偏移式组合式组合式组合, 以快速的离值计算方法展示了当前变式计算结果。