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An unconditionally stable solution using associated Hermite (AH) functions is proposed for the finite-difference time-domain (FDTD) method. The electromagnetic fields and their time derivatives in time-domain Maxwell's equations are expanded by these orthonormal basis functions. By applying Galerkin temporal testing procedure to these expanded equations the time variable can be eliminated from the calculations. A set of implicit equations is derived to calculate the magnetic filed expansion coefficients of all orders of AH functions for the temporal variable. And the electrical field coefficients can be obtained respectively. With the appropriate translation and scale parameters, we can find a minimum-order basis functions subspace to approach a particular electromagnetic field. The numerical results have shown that the proposed method can reduce the CPU time to 0.59% of the traditional FDTD method while maintaining good accuracy.

The time resolved three‐dimensional (3‐D) spatial reconstruction of lightning channels using high‐speed video (HSV) images and VHF broadband interferometer (BITF) data is first presented in this paper. Because VHF and optical radiations in step formation process occur with time separation no more than 1 μs, the observation data of BITF and HSV at two different sites provide the possibility of reconstructing the time resolved 3‐D channel of lightning. With the proposed procedures for 3‐D reconstruction of leader channels, dart leaders as well as stepped leaders with complex multiple branches can be well reconstructed. The differences between 2‐D speeds and 3‐D speeds of leader channels are analyzed by comparing the development of leader channels in 2‐D and 3‐D space. Since return stroke (RS) usually follows the path of previous leader channels, the 3‐D speeds of the return strokes are first estimated by combination with the 3‐D structure of the preceding leaders and HSV image sequences. For the fourth RS, the ratios of the 3‐D to 2‐D RS speeds increase with height, and the largest ratio of the 3‐D to 2‐D return stroke speeds can reach 2.03, which is larger than the result of triggered lightning reported by Idone. Since BITF can detect lightning radiation in a 360° view, correlated BITF and HSV observations increase the 3‐D detection probability than dual‐station HSV observations, which is helpful to obtain more events and deeper understanding of the lightning process.