## 3 Analysis of Accuracy and Performance
### Analysis of Accuracy  
In order to verify the stability of the proposed algorithm and eliminate the influence of various parameters in the ionosphere, we present a numerical simulation conducted in vacuum. To ensure the stability and dispersion of the algorithm, the simulation time is significantly extended, which can greatly verify the stability and dispersion of the algorithm. A sinusoidally modulated Gaussian pulse as an input electric-current profile is studied. The time dependence of the excitation function is as follows,  
\[S(t)=e^{\left(-\frac{(t-t_{0})^{2}}{\tau^{2}}\right)}\sin(2\pi f_{0}(t-t_{0})) \tag{48}\]  
Here, \(f_{0}\), \(t_{0}\), and \(\tau\) are constants. We choose \(f_{0}\) = 5 GHz and \(t_{0}=\tau=6.7\times 10^{-10}\) s. The total lattice dimension is 300 km \(\times\) 100 km \(\times\) 100 km. The current source is placed at the center of the domain and the observation point is placed 30 km away from the source. The computational domain is truncated by the UPML absorbing-boundary conditions. The electric-field component \(E_{r}\) at the observation point calculated by using the proposed HIE-FDTD method is plotted in Figure 4. For comparison, the results calculated by using the traditional FDTD method are also plotted in Figure 4.  
It can be seen from Figure 4 that the two results agree well with each other. The Gaussian pulse sources have limited duration and smooth waveforms, with a rapid increase in amplitude at the beginning and a rapid decrease to zero after reaching a peak. The simulation results are highly consistent with this theory, which indicates that the proposed method has excellent calculation accuracy. The simulation takes 27.1 min for the traditional FDTD method and 98.6 min for the proposed method. \begin{table}
\begin{tabular}{c c} \hline
**UPML Region** & **Parameter** \\ \hline Edge area & \(s_{r}=\kappa_{r}+\frac{\omega_{r}}{\mu\omega_{0}},\quad s_{\theta}=\kappa_{ \theta}+\frac{\omega_{\theta}}{\mu\omega_{0}}\) \\ Plane area & \(s_{r}=1,s_{\theta}=1\) \\ \hline \end{tabular}
\end{table} 
Table 1: The parameter settings for each region of the UPML model boundary. Figure 3: A simulation model with UPML. To further validate the accuracy and stability of the algorithm in the ionosphere, a pulsed signal with a center frequency of 20 kHz and a bandwidth of 300 Hz was adopted. The source was placed on the ground. The variation in the electric-field intensity at the observation point at a vertical distance of 0 to 100 km from the ground and the variation of the phase at the observation point at a horizontal distance of 0-200 km from the calculation source were calculated. This was used to study the calculation accuracy and precision of the algorithm proposed in this paper. Figure 5a,b give the comparison of daytime data and nighttime data of field intensity attenuation between the algorithm proposed in this paper and the traditional algorithm. From the two figures, the algorithm proposed in this paper and the traditional algorithm have a maximum difference of 2.3 dB in amplitude attenuation during the day and 4.2 dB at night. In this simulation, the CPU time of the traditional FDTD method is 124 min, while in the proposed HIE-FDTD method, it is 33 min, which is almost 1/4 times as that of the FDTD method. Obviously, the extremely large time step size applied in the proposed HIE-FDTD method results in the computational time being significantly reduced. Meanwhile, Figure 6a,b provide the comparison of daytime data and the nighttime of the phase between the algorithm proposed in this paper and the traditional algorithm. Figure 4: The electric-field values calculated by the FDTD algorithm and the proposed HIE-FDTD algorithm. Figure 5: (**a**) A comparison of the daytime data of field-intensity attenuation between the algorithm proposed in this paper and the traditional algorithm; (**b**) a comparison of the nighttime data of field-intensity attenuation between the algorithm proposed in this paper and the traditional algorithm.