# On the Lagged Diffusivity Method for the Solution of Nonlinear Finite Difference Systems
## 5 Numerical Experiments
### Blow-Up
#### 5.2.1 Preliminary Results and Analysis  
Let us first analyze how the computed solution evolves as the blow-up time approaches. To this end, let us start from \(t_{0}=0\) and see what happens at different times, until reaching \(T\); at first, we here  
Figure 4: Blow-up solutions and representation of where and how the blow-up occurs.  
choose \(\Delta t=0.1\) and \(N=100\). In Figure 5 we report what we get considering e.g., \(u_{2}^{*}\) (example of solutions with blow-up on one corner), \(u_{4}^{*}\) (example of solutions with blow-up on one or more edges) and \(u_{5}^{*}\) (example of solutions with blow-up on more corners). In Table 3 we report some information on errors and data of the iterative procedure.  
Of course, in all these cases, the iterative procedure stops when we are still quite far from the blow-up time: indeed, using \(\Delta t=0.1\), the last computable time step is at \(t=0.9\) when \(T=1\) and \(t=0.7\) when \(T=0.8\). Indeed, the blow-up occurs at \(t=0.9+\Delta t\) and at \(t=0.7+\Delta t\) respectively. We therefore need a smaller time step in order to get closer to the singularity. However, we notice that the computed solutions behave as the analytical ones (as proven by the errors in Table 3) and that, as the blow-up time approaches, they get more and more similar to the ones in Figure 4.  
It is also interesting to notice that the residual computed at the beginning of the lagged iteration, \(res_{0}\), gets larger as we get closer to \(T\). This can be viewed as an indicator of incoming blow-up: indeed, by the initialization \(\mathbf{u}^{(0)}=\mathbf{u}^{n}\), at the time step \(n+1\) we have \(res_{0}=\|\mathbf{F}(\mathbf{u}^{n})\|\). Therefore, \(res_{0}\) is the residual of \(\mathbf{F}(\mathbf{u})=0\) as in (9) for \(\mathbf{u}^{n}\) at time \(n+1\). So, if the solution is smooth in the time variable (and \(\Delta t\) is not excessively large), we do not expect \(res_{0}\) to be large: in these hypotheses, the solution \(\mathbf{u}^{n+1}\) will not be dramatically different from the solution \(\mathbf{u}^{n}\). However, if we have a singularity at some time \(T\), the solution vector will change faster and faster as we approach \(T\), leading to progressively larger initial residuals \(res_{0}\). In this light, the analysis of \(res_{0}\) gives information similar to the analysis of the derivative of \(\mathbf{u}^{*}\) with respect to \(t\).  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(\mathbf{t}\) & \(res_{0}\) & \(res\) & \(err_{h}\) & \(err_{2}\) rel. & \(\max\mathbf{u}\) & \(\min\mathbf{u}\) & \(k_{t}\) & \(j_{t}\) \\ \hline \multicolumn{8}{c}{\(u^{*}=u_{2}^{*}\), \(T=1.0\)} \\ \hline
0.3 & 369 & \(1.35\times 10^{-5}\) & \(1.74\times 10^{-5}\) & \(1.86\times 10^{-5}\) & 1.403 & 0.594 & 19 & 185 \\
0.6 & 1209 & \(8.66\times 10^{-6}\) & \(6.89\times 10^{-5}\) & \(6.27\times 10^{-5}\) & 2.405 & 0.721 & 21 & 202 \\
0.9 & 15,619 & \(1.57\times 10^{-5}\) & \(1.04\times 10^{-3}\) & \(6.88\times 10^{-4}\) & 8.402 & 0.918 & 25 & 254 \\ \hline \multicolumn{8}{c}{\(u^{*}=u_{4}^{*}\), \(T=0.8\)} \\ \hline
0.3 & 592 & \(1.03\times 10^{-5}\) & \(1.84\times 10^{-3}\) & \(1.88\times 10^{-3}\) & 1.930 & 0.505 & 20 & 186 \\
0.5 & 1828 & \(1.68\times 10^{-5}\) & \(1.76\times 10^{-3}\) & \(1.56\times 10^{-3}\) & 2.369 & 0.558 & 21 & 208 \\
0.7 & 22,484 & \(1.08\times 10^{-5}\) & \(1.54\times 10^{-2}\) & \(7.66\times 10^{-3}\) & 5.851 & 0.772 & 25 & 263 \\ \hline \multicolumn{8}{c}{\(u^{*}=u_{5}^{*}\), \(T=0.8\)} \\ \hline
0.3 & 112 & \(1.53\times 10^{-5}\) & \(3.51\times 10^{-4}\) & \(7.59\times 10^{-4}\) & 0.941 & 0.333 & 17 & 120 \\
0.5 & 346 & \(1.17\times 10^{-5}\) & \(3.29\times 10^{-4}\) & \(6.42\times 10^{-4}\) & 1.148 & 0.356 & 19 & 138 \\
0.7 & 3077 & \(1.45\times 10^{-5}\) & \(1.36\times 10^{-3}\) & \(1.85\times 10^{-3}\) & 2.694 & 0.433 & 22 & 175 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Error and data of the lagged diffusivity procedure for \(u_{2}^{*}\), \(u_{4}^{*}\) and \(u_{5}^{*}\) as \(t\) approaches \(T\).