## 5 Numerical Results  
In this section, the effectiveness of the proposed method is shown. The convergence rate is examined by means of the following truncation error:  
\[\textit{err}_{N}(M)=\sqrt{\frac{\sum_{n=-N+1}^{N-1}\left\|\textbf{x}_{M+1}^{( n)}-\textbf{x}_{M}^{(n)}\right\|^{2}}{\sum_{n=-N+1}^{N-1}\left\|\textbf{x}_{M}^{(n)} \right\|^{2}}}, \tag{25}\]  
where the symbol \(\left\|\cdot\right\|\) is used to denote the Euclidean norm, \(\textbf{x}_{M}^{(n)}\) is the vector of the first \(M\) expansion coefficients for each unknown, and \(2N-1\) is the number of cylindricalharmonics considered. For each example, together with the behavior of the truncation error, a near-field parameter, i.e., the amplitude of the surface current density on the resistive disk, and a far-field parameter, i.e., the BRCS, are reconstructed. It is worth observing that the number of cylindrical harmonics to be used are estimated by means of the formula provided in [58], and the symmetries of the coefficient matrix allow the reduction in the number of elements to be numerically evaluated to \(NM(2M+1)\). As a result, only a few seconds are needed to accurately reconstruct the solution in all the examples proposed by using an in-house software code implemented in a C++ environment on a laptop equipped with an Intel Core i7-10510U 1.8 GHz, 16 GB RAM. In Figure 2, the truncation error for varying values of \(M\), the amplitude of the surface current density on the disk along the direction \(\theta=90^{\circ}\) and \(\phi=0^{\circ},180^{\circ}\), and the BRCS in the plane \(\phi=0^{\circ},180^{\circ}\) are plotted for \(a=2\lambda\), \(R_{e}=1\) k\(\Omega\) and different values of the resistivity of the disk, i.e., \(R_{i}=10\)\(\Omega,100\)\(\Omega,500\)\(\Omega\), when a plane wave with TE polarization with respect to the \(z\)-axis impinges onto the structure with \(\theta_{0}=30^{\circ}\) and \(\phi_{0}=0^{\circ}\). The convergence is very fast in all the cases examined in terms of numbers of cylindrical harmonics and expansion functions to be used in order to achieve a prescribed accuracy. Indeed, \(N=15\) is considered for all the cases and, in the worst case (\(R_{i}=10\)\(\Omega\)), the error reduced below \(10^{-1},10^{-2},10^{-3}\) for \(M=4,9,21\), respectively. It is interesting to observe that the convergence is faster the closer the values of the resistivities \(R_{i}\) and \(R_{e}\) are, i.e., the weaker the discontinuity is. Of course, even the amplitude of the diffracted field and then, the BRCS are smaller the weaker the discontinuity is. On the other hand, the decreasing amplitude of the surface current density for increasing values of \(R_{i}\) can be explained by the higher transparency associated with higher values of \(R_{i}\). To conclude, as expected, the BRCS shows two maxima, in the forward direction and in the specular with respect to the incidence direction. In Figure 3, the same parameters as in Figure 2 are plotted for \(R_{i}=100\)\(\Omega\), \(R_{e}=1\) k\(\Omega\), and different values of the disk radius, i.e., \(a=\lambda,\ 2\lambda,\ 4\lambda\), when a TM polarized plane wave impinges onto the structure with \(\theta_{0}=30^{\circ}\) and \(\phi_{0}=0^{\circ}\). Of course, the number of cylindrical harmonics and expansion functions needed to achieve a given accuracy increases by increasing the radius of the disk. Moreover, the convergence is very fast in all the cases examined. Indeed, \(N=11,\ 15,\ 23\) for \(a=\lambda,\ 2\lambda,\ 4\lambda\), respectively. Moreover, in the worst case (\(a=4\lambda\)), errors smaller than \(10^{-1},10^{-2},10^{-3}\) are achieved for \(M=7,\ 11,\ 16,\) respectively. As expected, the surface current density and the BRCS show an increasing number of oscillations for increasing values of the disk radius. In Figure 4, in order to validate the implemented in-house software code, comparisons with CST-MWS are presented.