## 2 Formulation of the Hybrid Method
### Scattered Field Generated by the DGTD Region
\begin{array}{ll}J_{sk,qt}=k_{h}^{m}\cancel{ff}_{\partial\tau_{m}}I_{s }\cdot N_{qt}^{m}ds,&J_{sv,qt}=v_{e}^{m}\cancel{ff}_{\partial\tau_{m}}(n\times I _{s})\cdot N_{qt}^{m}ds\\ M_{sk,qt}=k_{e}^{m}\cancel{ff}_{\partial\tau_{m}}M_{s}\cdot N_{qt}^{m}ds,&M_{ sv,qt}=v_{e}^{m}\cancel{ff}_{\partial\tau_{m}}(n\times M_{s})\cdot N_{qt}^{m}ds \end{array} \right. \tag{6}\]  
where \(\{M\}\) is the mass matrix; \(\{S\}\) is the stiffness matrix; \(\Big{\{}F_{kh}^{k}\Big{\}}\), \(\{F_{kh}\}\), \(\Big{\{}G_{ve}^{k}\Big{\}}\) and \(\{G_{ve}\}\) are the flux matrices. \(E_{m}\) and \(H_{m}\), and \(E_{k}\) and \(H_{k}\) are the edge electric field and magnetic field of the \(m\)-th element. The superscript \(k\) represents adjacent elements of \(m\) elements. The above matrix \(\{M\}\) and \(\{S\}\) elements are:  
\[M_{qt,q}=\cancel{ffff}_{\tau_{m}}N_{qt}^{m}\cdot N_{q}^{m}dv,\qquad S_{qt^{ \prime},q}=\cancel{ffff}_{\tau_{m}}N_{qt^{\prime}}^{m}\cdot\Big{(}\
abla\times N _{q}^{m}\Big{)}dv,\quad q,qt=1,2,\ldots,6 \tag{7}\]  
and flux matrices \(\Big{\{}F_{kh}^{k}\Big{\}}\), \(\{F_{kh}\}\), \(\Big{\{}G_{ve}^{k}\Big{\}}\) and \(\{G_{ve}\}\) are:  
\[\left\{ \begin{array}{ll}F_{kh,qt}=k_{h}^{m}\cancel{ff}_{\partial\tau_{m}}N_ {qt}^{m}\cdot\Big{(}n\times N_{q}^{m}\Big{)}ds,\quad F_{k,qt}=k_{e}^{m}\cancel{ ff}_{\partial\tau_{m}}N_{qt}^{m}\cdot\Big{(}n\times N_{q}^{m}\Big{)}ds\\ G_{vh,qt}=v_{h}^{m}\cancel{ff}_{\partial\tau_{m}}N_{qt^{\prime}}^{m}\cdot \Big{(}n\times n\times N_{q}^{m}\Big{)}ds,\quad G_{ve,qt}=v_{e}^{m}\cancel{ ff}_{\partial\tau_{m}}N_{qt^{\prime}}^{m}\cdot\Big{(}n\times n\times N_{q}^{m}\Big{)}ds\\ F_{kh,qt}^{k}=k_{h}^{m}\cancel{ff}_{\partial\tau_{m}}N_{qt^{\prime}}^{m}\cdot \Big{(}n\times N_{p}^{m+}\Big{)}ds,\quad F_{k,qt}^{k}=k_{e}^{m}\cancel{ff}_{ \partial\tau_{m}}N_{qt}^{m}\cdot\Big{(}n\times N_{p}^{m+}\Big{)}ds\\ G_{vh,qt}^{k}=v_{h}^{m}\cancel{ff}_{\partial\tau_{m}}N_{qt^{\prime}}^{m}\cdot \Big{(}n\times n\times N_{p}^{m+}\Big{)}ds,\quad G_{ve,qt}^{k}=v_{e}^{m}\cancel{ ff}_{\partial\tau_{m}}N_{qt^{\prime}}^{m}\cdot\Big{(}n\times n\times N_{p}^{m+}\Big{)}ds \end{array} \right. \tag{8}\]  
where \(q,qt,p=1,2,\ldots,6\), and \(N_{qt^{\prime}}^{m+}\) is the m-th adjacent element basis function. In this paper, the hybrid explicit-implicit time integration method [7] is used to calculate the multi-scale problems. In this method, the implicit time integration scheme is locally applied in the fine region of the grid, while the explicit time scheme is retained in the complementary part, which can further improve the stability of the calculation and reduce the amount of calculation. As shown in Figure 3, the perfect match layer (PML) is used as boundary condition. We utilize a time-domain near-field to far-field (NFFF) transformation method from [36], and calculate the scattered electric and magnetic fields \(\cancel{zz}_{\rm DGTD}(r,t)\) and \(\cancel{zz}_{\rm DGTD}(r,t)\) in far field. In order to compute the NFFF transformation, a proper Huygens surface is considered to be in the DGTD region, and then, the time-domain scattering field in DGTD region can be obtained. Figure 3: DGTD region.