## 3 Results and Discussion
### Lightning-Induced Phase Conductor Current  
The aim of this section was to consider a transient problem of practical relevance, from the perspective of both transmission line modeling and pipeline integrity. [PERSON] and colleagues used a finite-difference time domain methodology in [49] to simulate the effects of a lightning strike on a \(500\,\mathrm{kV}\) transmission line tower. The authors computed the lightning-induced current on one of the phase conductors. We assumed that the same current flows through phase 1 of the transmission line described in Section 3.1, and we used two different numerical strategies to assess the transient response of the whole system \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Region**} & \multicolumn{3}{c}{**Current Magnitude (A)**} & \multicolumn{3}{c}{**Phase (deg)**} \\ \cline{2-7}  & **FLARE** & **COMSOL** & \(\mathbf{\Delta_{r}\%}\)1 ** & **FLARE** & **COMSOL** & \(\mathbf{\Delta_{r}\%}\)1 ** \\ \hline OGW 1 & 12.2731 & 12.2730 & 0.0004 & \(-140.8317\) & \(-140.9354\) & 0.0736 \\ OGW 2 & 10.2212 & 10.2205 & 0.0074 & 42.3653 & 42.2765 & 0.2100 \\ pipe & 10.4912 & 10.4877 & 0.0332 & \(-10.7012\) & \(-10.7105\) & 0.0864 \\ Soil & 8.1174 & 8.1091 & 0.1023 & 166.3202 & 166.3265 & 0.0038 \\ \hline \end{tabular}
\end{table} 
Table 1: Time-harmonic case (60 Hz)—comparison between the currents computed with FLARE and COMSOL Multiphysics®. Figure 4: Magnitude of the MVP \(z\)- component \(A_{z}(x,y)\) in \(\mathrm{T}\,\mathrm{m}^{-1}\) and magnetic field lines when \(I_{\text{phase 1}}=250\,\mathrm{A}\), \(t=1\times 10^{-3}\,\mathrm{s}\). The OGWs and the pipeline are not to scale. under these conditions. We used a Heidler function [52] to represent the phase 1 current from [49], i.e.,  
\[i_{\rm phase\,1}(t)=-\frac{I_{0}}{\eta}\frac{(t/\tau_{1})^{n}}{1+(t/\tau_{1})^{n }}\exp\biggl{(}-\frac{t}{\tau_{2}}\biggr{)}, \tag{20}\]  
where \(I_{0}=1\times 10^{4}\) A, \(\tau_{1}=1.25\times 10^{-6}\) s, \(\tau_{2}=5.5\times 10^{-5}\) s, \(\eta=0.875\), and \(n=2\). We started by simulating the transient response of the system to \(i_{\rm phase\,1}(t)\) with the time-domain solver in FLARE and COMSOL Multiphysics(r) over 1 ms. Concerning FLARE, the automatic time-stepping procedure described in Section 2.1.3 led to 158 time steps. The time step lengths ranged from \(\Delta t\approx 1\times 10^{-13}\) s at the beginning of the simulation when \(i_{\rm phase\,1}(t)\) increased rapidly to \(\Delta t\approx 1\times 10^{-4}\) s in the latter stages of the \(i_{\rm phase\,1}(t)\) decay. The total execution time (wall-clock time) of the simulation, including the postprocessing, was \(t_{\rm WC}\approx 135\) s on the CPU used for testing (AMD Ryzen 7 PRO 4750U). In comparison, COMSOL Multiphysics(r) (which also uses an adaptive time-stepping technique) performed the simulation with approximately half the time steps, requiring \(\approx\)45 s.  
The time evolution of the currents computed with FLARE is shown in Figure 5 over a limited portion of the simulation time. The gray continuous line represents \(i_{\rm phase\,1}(t)\), enforced on phase 1. The square markers on \(i_{\rm phase\,1}(t)\) mark the time instants used for the computation. All the obtained currents were positive, in contrast to \(i_{\rm phase\,1}(t)\). While the pipeline had a lower electrical conductivity with respect to the phase conductors and was partially screened by the soil, its maximum induced current was larger than the one in phase 3. This was due to a combination of its location in close proximity to the transmission line and its surface, considerably larger than the one of either the phase or overhead ground conductors. In order to verify the accuracy of the solution obtained with FLARE, we compared a subset of the currents shown in Figure 5 to the same quantities evaluated over time using the COMSOL Multiphysics(r) software. We also highlight that the FLARE wall-clock time was comparable or lower to that needed to perform the same simulation in COMSOL Multiphysics(r) on the same machine.