# Enhancing the Iterative Smoothed Particle Hydrodynamics Method
## 4 Numerical Results  
The [PERSON]'s function (13) with \(d=2\) depicted in Figure 1, taken from the scattered data literature [23, 24], is referred to as test function  
\[ \begin{array}{ll}f(x^{(1)},x^{(2)})=&0.75\exp\big{[}-\frac{(9x^{(1)}-2)^{2} +(9x^{(2)}-2)^{2}}{4}\big{]}+\\ &+0.75\exp\big{[}-\frac{(9x^{(1)}+1)^{2}}{49}-\frac{(9x^{(2)}+1)^{2}}{10}\big{]} +\\ &+0.5\exp\big{[}-\frac{(9x^{(1)}-7)^{2}+(9x^{(2)}-3)^{2}}{4}\big{]}+\\ &-0.2\exp\big{[}-(9x^{(1)}-4)^{2}-(9x^{(2)}-7)^{2}\big{]}.\end{array}  \tag{13}\]  
Two sets of data sites are considered: the gridded \(\Xi_{G}\), composed of regular distribution points, and the Halton \(\Xi_{H}\)[25] which consists of unevenly distribution points. The _haltonset_ function of MATLAB\(\copyright\) is used to generate \(\Xi_{H}\). The accuracy of the estimates is measured with  
\[RMSE=\sqrt{\frac{\sum_{i=1}^{M}|f_{h}(\mathbf{x}_{i})-f(\mathbf{x}_{i})|^{2}}{ M}} \tag{14}\]  
by fixing \(M=1600\) as the number of evaluation points and \(N=(2^{t}+1)^{2}\), \(t=1\),2,.., as the number of data sites increasing in the unit domain. The kernel function is the Gaussian one. Some results on the accuracy and on the convergence are illustrated. The improvements in the linear approximation are evident in Table 1, where the RMSEs for the standard and the formulation with \(p=1\) are included, referring to the \(\Xi_{G}\) and \(\Xi_{H}\) data distributions. In Tables 2 and 3, the error behavior adopting the iterative process ISPH, assuming \(f_{h}^{(0)}\) with \(p=1\), is shown for \(\Xi_{G}\) and \(\Xi_{H}\) respectively. A significant reduction, for the two data sequences, can be appreciated by increasing \(N\) and the number of the iterations. In Tables 4 and 5, the RMSEs refer to the enhanced iterative approach E-ISPH for \(\Xi_{G}\) and \(\Xi_{H}\) showing an accuracy comparable to ISPH requiring a reduced number of iterations. In Figures 2-4 the RMSEs behavior is reported in a loglog plot comparing the SPH,ISPH and E-ISPH approaches. In Figure 5, the time (s), required by the ISPH and E-ISPH to reach a comparable accuracy in the approximation, is exhibited in a loglog plot too. All simulations have been executed with a processor 2.3 GHz Intel Core i9 8 core. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\mathbf{N}\) & \(f_{h}^{(0)}\) & \(\tilde{f}_{h}^{(10)}\) & \(\tilde{f}_{h}^{(20)}\) & \(\tilde{f}_{h}^{(30)}\) & \(\tilde{f}_{h}^{(40)}\) \\ \hline
9 & 0.2624 & 0.1312 & 0.3210 & 0.3210 & 0.3210 \\
25 & 0.2206 & 0.0861 & 0.2719 & 0.3173 & 0.3173 \\
81 & 0.1244 & 0.0172 & 0.0545 & 0.0435 & 0.0446 \\
289 & 0.0693 & 0.0023 & 0.0011 & 0.0009 & 0.0013 \\
1089 & 0.0266 & \(6.95\times 10^{-5}\) & \(2.64\times 10^{-6}\) & \(9.78\times 10^{-7}\) & \(7.43\times 10^{-5}\) \\
4225 & 0.0137 & \(3.13\times 10^{-5}\) & \(1.74\times 10^{-7}\) & \(1.31\times 10^{-8}\) & \(3.17\times 10^{-5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: RMSEs for the E-ISPH, \(f_{h}^{(0)}\) with \(p=1\). Data sites \(\Xi_{H}\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\mathbf{N}\) & \(f_{h}^{(0)}\) & \(\tilde{f}_{h}^{(10)}\) & \(\tilde{f}_{h}^{(100)}\) & \(\tilde{f}_{h}^{(100)}\) \\ \hline
9 & 0.2624 & 0.1626 & 0.1427 & 0.1280 \\
25 & 0.2206 & 0.1331 & 0.1023 & 0.0839 \\
81 & 0.1244 & 0.0603 & 0.0269 & 0.0167 \\
289 & 0.0693 & 0.0133 & 0.0042 & 0.0021 \\
1089 & 0.0266 & 0.0017 & \(2.35\times 10^{-4}\) & \(4.60\times 10^{-5}\) \\
4225 & 0.0137 & \(9.15\times 10^{-4}\) & \(1.27\times 10^{-4}\) & \(1.71\times 10^{-5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: RMSEs for the ISPH, \(f_{h}^{(0)}\) with \(p=1\). Data sites \(\Xi_{H}\). Figure 2: _Cont._Figure 3: RMSEs versus number of data sites (\(\Xi_{H}\)). (**a**) Comparison between SPH and ISPH; (**b**) comparison between SPH and E-ISPH. All the computations are with starting normalized values and \(10\leq n\leq 40\). Figure 2: RMSEs versus number of data sites (\(\Xi_{G}\)). (**a**) Comparison between SPH and ISPH; (**b**) comparison between SPH and E-ISPH.