## 3 Results and Discussion
### Numerical Validation of the Inverse-Laplace Approach
For this reason we provide our own implementation of a convolution integral routine, iw=intconv(tiw,tu,u,tv,). The intconv function requires as input the array of the two signals to be combined, u and v, together with their respective temporal indices tu and tv. The two input signals are interpolated internally and the convolution integral iw is then returned at times specified with the array tiw. In order to validate this proposed methodology, we compared the pipeline current computed with this approach to the solution of the implicit time-stepping algorithm discussed in the previous section. The current impulse response of the pipeline \(h_{\text{pipe}}(t)\) was computed over 250 instants, logarithmically spaced between \(1\times 10^{-11}\) s and 10 s. Figure 7 shows the time-stepping results (black continuous line) compared to a subset of the values obtained using the convolution integral. The agreement between the two approaches was satisfactory over the whole time interval covered by the simulation. Focusing on the region with the largest discrepancy between the two approaches, highlighted in Figure 7, the pipeline current values yielded by the two approaches were 811.899 A (convolution) and 811.203 A (time domain). An even closer agreement could be obtained by employing a finer time grid for the pipeline impulse response calculation with the numerical inverse-Laplace approach. We now want to briefly comment on the efficiency of the two approaches used in this work for transient problems, i.e., time-marching and the discussed inverse-Laplace technique. For the specific transient problem considered in this work, the time-marching analysis in FLARE (with a variable time step) required computation times between one and two minutes (depending on the initial time step length and sensitivity of the adaptive time step control). Comparing this performance to the ones of the inverse-Laplace approach is not trivial and is strongly dependent on the specific use case. The picture also changes depending on how the inverse-Laplace approach is used, i.e., if (1) the considered variable is directly anti-transformed or if (2) the inverse-Laplace approach is used to compute an impulse response of the system, to be later convoluted with a given signal. The number of function evaluations in the Laplace domain (i.e., solutions of a stationary problem for a single value of \(s\)) required to anti-transform the solution at a given time \(t\) may change considerably (depending on the source terms and the specific value of \(t\)) when using the adaptive procedure by [PERSON] et al. Therefore, while 10 or 20 computations might be sufficient for a given \(t=\tau_{1}\), up to 100 or 200 evaluations might be required for \(t=\tau_{2}\gg\tau_{1}\). Still, as we will see in a later section, a totally different situation is obtained when using  
Figure 7: Comparison between the pipeline current computed with the implicit time-marching algorithm (time domain) and inverse-Laplace approaches (convolution) in FLARE. We denote the results of the inverse-Laplace as _convolution_, since the pipeline current is obtained by convolving the inverse-Laplace transform of the impulse response of the system and the lightning-induced current on phase 1.  
the Talbot and Euler algorithms. In any case, one of the advantages of the inverse-Laplace approach is that the problem can be massively parallelized--as we implemented in FLARE--given that the results at different time instants are independent. As a rule of thumb, if the inverse-Laplace is directly used without computing an impulse response (approach (1)), we can say that the time-marching approach is generally more efficient if used to observe the behavior of a quantity over large time-spans using a correspondingly large number of sampling points. In this case, the inverse-Laplace approach can still be competitive if the solution is just needed for a few key instants that can be computed in parallel. The situation changes if--as we implemented in this work--the above-described approach (2) is used.